HighSchoolQ

➕ Algebra 1 · Factoring

Difference of Squares

Factor a difference of squares into (a + b)(a − b), spot perfect squares, take out the GCF first, and see why a sum like x² + 9 does not factor.

Grades 9, 10⏱ 15 min✔ AI-checkedCCSS A-SSE.A.2
About standards and checking

Written by AI, then independently re-solved and checked by separate AI reviewers and automated answer checks. Written by Claude (Sonnet 5.5) from the concept template. Automated gate: schema, notation and every math answer verified by evaluated expressions. Two blind re-solves by separate AI solvers with no access to the answer key: every keyed answer matched. Two prose audits by a separate, stronger AI reviewer (Claude Opus 5.5): every finding fixed, no wrong answer keys found. Final small edits re-checked by the automated gate (no human teacher review yet). Last checked 2026-11-08.

  • A-SSE.A.2: Use the structure of an expression to identify ways to rewrite it.

By the end you can…

  • Recognize a difference of squares: a binomial with one perfect square subtracted from another, including terms like 9y2 and x6.
  • Factor it with a2 − b2 = (a + b)(a − b), and check the answer by expanding.
  • Take out a greatest common factor first, and keep factoring until the expression is completely factored, as in x4 − 16.
  • Explain why a sum of squares such as x2 + 9 does not factor, and use the pattern for mental math such as 51 × 49.

Lesson (school style)

A difference of squares is a binomial in which one perfect square is subtracted from another, like x2 − 9. It factors in one step: a2 − b2 = (a + b)(a − b). Factoring is the reverse of multiplying, so you can always check an answer by multiplying it back out. In this lesson, “factor” means “factor using integers,” and the pattern needs a minus sign between two perfect squares.

The pattern

a2 − b2 = (a + b)(a − b)

The two factors use the same two pieces, a and b. They differ only in the sign between them: one plus and one minus. The order of the factors does not matter, because multiplication can be done in either order.

Try it with numbers. For a = 9 and b = 4: a2 − b2 = 81 − 16 = 65, and (9 + 4)(9 − 4) = 13 × 5 = 65.

The letters a and b can stand for anything you can square: numbers, variables, or terms such as 2x. In x2 − 9 we have a = x and b = 3, so x2 − 9 = (x + 3)(x − 3).

If you met the product of a sum and a difference in Special Products of Binomials, this is that same fact read from right to left.

Why it works: expand and look at the middle

Multiply (a + b)(a − b) one term at a time:

(a + b)(a − b) = a·a − a·b + b·a − b·b = a2 − ab + ab − b2 = a2 − b2

The two middle terms, −ab and +ab, have the same size and opposite signs, so they cancel. They cancel only because the two factors have opposite signs. In (a + b)(a + b) the middle terms are both +ab and add up to 2ab, so (a + b)2 = a2 + 2ab + b2. That is not a difference of squares.

Why it works: a picture

Start with a big square whose sides are a long. Cut a small square with sides b long out of one corner. (In the picture, a is bigger than b.) The L-shaped piece that is left has area a2 − b2.

Cut the L into two rectangles: a bottom strip that is a wide and a − b tall, and a piece above it that is a − b wide and b tall. Turn the second piece on its side and slide it to the end of the strip. Together they make one rectangle that is a + b wide and a − b tall. Its area is (a + b)(a − b).

Nothing was added or thrown away, so the two areas are equal: a2 − b2 = (a + b)(a − b).

The picture needs positive lengths with a > b. The expanding argument above shows the identity is true for all numbers.

Spotting perfect squares

A perfect square is what you get when you square something. With whole-number coefficients, the numbers to know are the squares of 1 through 12: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121 and 144.

A term is a perfect square when both of these are true:

  • its coefficient is a perfect square (a lone variable power such as x6 has coefficient 1, which counts), and
  • every variable in it has an even exponent.

To find the thing that was squared, take the square root of the coefficient and halve each exponent. Halving works because squaring doubles an exponent: (x3)2 = x3 · x3 = x6.

  • 1 = 12
  • 49 = 72
  • x6 = (x3)2
  • 9y2 = (3y)2
  • 16x4 = (4x2)2

Not perfect squares: 20x2 (20 is not a perfect square), x5 (the exponent is odd) and 7.

Put it together with 4x2 − 25: 4x2 = (2x)2 and 25 = 52, so a = 2x and b = 5. Then 4x2 − 25 = (2x + 5)(2x − 5). Check by expanding: 4x2 − 10x + 10x − 25 = 4x2 − 25.

The steps

  1. Take out any GCF (greatest common factor) of all the terms first.
  2. Check the shape. You need exactly two terms, and each must be a perfect square, with one subtracted from the other. It does not matter which square is written first: 100 − 9t2 qualifies.
  3. Find a and b, the things that were squared.
  4. Write (a + b)(a − b).
  5. Check by expanding. The middle terms should cancel and you should land back on the start.
  6. Look again. If any factor can still be factored, keep going until the expression is completely factored.

Take out the GCF first

Sometimes the squares are hiding behind a common factor. In 20x2 − 5, neither 20 nor 5 is a perfect square, so it does not look like a difference of squares. But 5 divides both terms: 20x2 − 5 = 5(4x2 − 1). Now 4x2 = (2x)2 and 1 = 12, so 20x2 − 5 = 5(2x + 1)(2x − 1). The GCF stays out front of the finished product.

Looking for a GCF first is a strategy, not a guarantee. After you take it out, the terms still have to be perfect squares. For 3x2 − 20 the GCF is 1 and neither term is a perfect square, so this pattern does not apply.

Completely factored: keep going

An expression is completely factored when you have taken out any GCF and every other factor is prime, meaning it cannot be factored any further using integers.

Look at x4 − 16. Since x4 = (x2)2 and 16 = 42, it is a difference of squares with a = x2 and b = 4:

x4 − 16 = (x2 + 4)(x2 − 4)

Seeing x4 − 16 as (x2)2 − 42 means reading the structure of the expression: the same pattern can sit inside bigger pieces.

Now inspect each factor. The factor x2 − 4 = x2 − 22 is another difference of squares, so it factors again: (x + 2)(x − 2). The factor x2 + 4 is a sum of squares and cannot be factored (see the next section). So

x4 − 16 = (x2 + 4)(x + 2)(x − 2)

Stopping at (x2 + 4)(x2 − 4) is a correct factorization, but it is not complete, because x2 − 4 can still be factored.

Why a sum like x² + 9 does not factor

Compare x2 − 9 and x2 + 9. The first is (x + 3)(x − 3). The second is a sum of squares, and the pattern needs a minus sign.

Try to force it. (x + 3)(x − 3) = x2 − 9 has the wrong sign. (x + 3)(x + 3) = x2 + 6x + 9 and (x − 3)(x − 3) = x2 − 6x + 9 have a middle term. To make the middle terms cancel, the signs must be opposite, and opposite signs make the last term negative. So no choice of signs gives x2 + 9.

That only tried factors that look like (x ± 3). Here is an argument for all numbers. Any way of writing x2 + 9 as a product of two binomials with real coefficients can be put in the form (x + p)(x + q), because the x2 term has coefficient 1. Put x = −p. The right side becomes 0. The left side becomes p2 + 9, which is at least 9, never 0. So no real numbers p and q work. The same idea works for any sum like 4x2 + 25.

So x2 + 9 is prime: it cannot be factored using integers, and it cannot be factored with any real-number coefficients either. (A later course uses imaginary numbers, with i2 = −1, to write x2 + 9 = (x + 3i)(x − 3i).)

A sum of squares can still have a GCF: 4x2 + 36 = 4(x2 + 9). The GCF 4 comes out, and the x2 + 9 stays as it is.

A caution about how far this goes: this section is about two-term sums of degree 2, like x2 + 9 and 4x2 + 25. A few sums with higher powers, such as x4 + 4 = (x2 + 2x + 2)(x2 − 2x + 2), can be factored by methods beyond this lesson.

Mental math

The pattern also multiplies numbers quickly. If two numbers are the same distance from a number whose square you know, one is a sum and the other is a difference: (n + d)(n − d) = n2 − d2.

For 47 × 53: both numbers are 3 away from 50, so 47 × 53 = (50 − 3)(50 + 3) = 502 − 32 = 2500 − 9 = 2491.

It works only when the numbers sit the same distance on either side of the middle number. A pair like 47 × 52 is not the same distance from 50, so the trick does not apply with 50 as the middle.

Key words

Difference of squares
A binomial in which one perfect square is subtracted from another, such as x2 − 9 or 100 − 9t2. It factors as a2 − b2 = (a + b)(a − b).
Perfect square
A term that is another term squared. With whole-number coefficients, the coefficient must be 1, 4, 9, 16, 25, … and every variable exponent must be even. Examples: 49, 9y2 (which is (3y)2) and x6 (which is (x3)2).
Sum of squares
Two perfect squares added, such as x2 + 9. After any GCF is taken out, a two-term sum of squares of degree 2 is prime.
Binomial
A polynomial with exactly two terms, such as x2 − 9.
Greatest common factor (GCF)
The largest factor that divides every term. For 18x2 − 50 it is 2.
Prime (cannot be factored using integers)
A polynomial that cannot be written as a product of simpler polynomials with integer coefficients, other than by taking out 1 or −1. The polynomial x2 + 9 is prime. The polynomial 2x + 4 is not, because 2x + 4 = 2(x + 2).
Completely factored
Written as a product of any GCF and prime factors, so that no factor can be factored further using integers. For x4 − 16 that is (x2 + 4)(x + 2)(x − 2).

Worked examples

Example 1. Factor x2 − 64.

  1. Check the shape: there are two terms, a minus sign between them, and both are perfect squares. x2 is x squared and 64 = 82. There is no GCF other than 1.
  2. So a = x and b = 8.
  3. Write (a + b)(a − b) = (x + 8)(x − 8).
  4. Check by expanding: (x + 8)(x − 8) = x2 − 8x + 8x − 64 = x2 − 64. It matches the start.

Answer: Answer: (x + 8)(x − 8)

Example 2. Factor 49x2 − 16.

  1. 49x2 = (7x)2 and 16 = 42, so a = 7x and b = 4. The coefficient gets a square root too: 49 becomes 7.
  2. Write (7x + 4)(7x − 4).
  3. Check by expanding: (7x + 4)(7x − 4) = 49x2 − 28x + 28x − 16 = 49x2 − 16.

Answer: Answer: (7x + 4)(7x − 4)

Example 3. Factor x6 − 121.

  1. Halve the exponent: x6 = (x3)2, because (x3)2 = x3 · x3 = x6. Also 121 = 112.
  2. So a = x3 and b = 11, and the factorization is (x3 + 11)(x3 − 11).
  3. Check by expanding: (x3 + 11)(x3 − 11) = x6 − 11x3 + 11x3 − 121 = x6 − 121.
  4. Neither factor is a difference of squares, because 3 is an odd exponent and 11 is not a perfect square. They are in fact prime, but proving it, and factoring cubes like x3 − 8 in other problems, takes tools from a later course.

Answer: Answer: (x3 + 11)(x3 − 11)

Example 4. Factor 100 − 9t2 (the number comes first this time).

  1. Both terms are perfect squares: 100 = 102 and 9t2 = (3t)2. The square being subtracted is 9t2. It does not matter which square comes first.
  2. So a = 10 and b = 3t, and the factorization is (10 + 3t)(10 − 3t).
  3. Check by expanding: (10 + 3t)(10 − 3t) = 100 − 30t + 30t − 9t2 = 100 − 9t2.
  4. Writing it as −(3t + 10)(3t − 10) gives the same expression, because −(9t2 − 100) = 100 − 9t2.

Answer: Answer: (10 + 3t)(10 − 3t)

Example 5. Factor x4 − 16 completely.

  1. x4 = (x2)2 and 16 = 42, so a = x2 and b = 4: x4 − 16 = (x2 + 4)(x2 − 4).
  2. Inspect each factor. x2 + 4 is a sum of squares, so it is prime and stays.
  3. x2 − 4 = x2 − 22 is a difference of squares, so it factors again: (x + 2)(x − 2).
  4. Put the pieces together: (x2 + 4)(x + 2)(x − 2).
  5. Check by expanding in two stages: (x + 2)(x − 2) = x2 − 4, and then (x2 + 4)(x2 − 4) = x4 − 4x2 + 4x2 − 16 = x4 − 16.

Answer: Answer: (x2 + 4)(x + 2)(x − 2)

Example 6. Factor 18x2 − 50 completely.

  1. Neither 18 nor 50 is a perfect square, so look for a GCF first. Both are divisible by 2, and no larger number divides both: 18x2 − 50 = 2(9x2 − 25).
  2. Now 9x2 = (3x)2 and 25 = 52, so 9x2 − 25 = (3x + 5)(3x − 5).
  3. Keep the 2 out front: 2(3x + 5)(3x − 5).
  4. Check by expanding: (3x + 5)(3x − 5) = 9x2 − 25, and 2(9x2 − 25) = 18x2 − 50.

Answer: Answer: 2(3x + 5)(3x − 5)

Example 7. Factor 5x4 − 5 completely.

  1. The GCF is 5: 5x4 − 5 = 5(x4 − 1).
  2. x4 = (x2)2 and 1 = 12, so x4 − 1 = (x2 + 1)(x2 − 1).
  3. The factor x2 − 1 = x2 − 12 factors again as (x + 1)(x − 1). The factor x2 + 1 is a sum of squares, so it stays.
  4. Together: 5(x2 + 1)(x + 1)(x − 1).
  5. Check by expanding: (x + 1)(x − 1) = x2 − 1, then (x2 + 1)(x2 − 1) = x4 − 1, and 5(x4 − 1) = 5x4 − 5.

Answer: Answer: 5(x2 + 1)(x + 1)(x − 1)

Example 8. Which of these can be factored with the difference-of-squares pattern using integers? (a) 9x2 − 1 (b) x2 + 49 (c) x2 − 10 (d) 36 − x4. Factor the ones that can.

  1. (a) 9x2 = (3x)2 and 1 = 12, with a minus sign between them. Yes: 9x2 − 1 = (3x + 1)(3x − 1).
  2. (b) There is a plus sign, so this is a sum of squares. No: x2 + 49 is prime.
  3. (c) The terms are x2 and 10, and 10 is not a perfect square. No. (If square roots were allowed it would be (x + √10)(x − √10), but those are not integers.)
  4. (d) 36 = 62 and x4 = (x2)2, with the square being subtracted written second. Yes: 36 − x4 = (6 + x2)(6 − x2). Neither factor can be factored further with integers: 6 + x2 is a sum, and 6 − x2 would need 6 to be a perfect square.
  5. Check (a): (3x + 1)(3x − 1) = 9x2 − 1. Check (d): (6 + x2)(6 − x2) = 36 − x4.

Answer: Answer: (a) (3x + 1)(3x − 1). (b) No, it is prime. (c) No. (d) (6 + x2)(6 − x2).

Example 9. Use the pattern to find 51 × 49 in your head.

  1. Both numbers are 1 away from 50: 51 = 50 + 1 and 49 = 50 − 1.
  2. So 51 × 49 = (50 + 1)(50 − 1) = 502 − 12.
  3. 502 = 2500, so 2500 − 1 = 2499.
  4. Check the long way: 51 × 49 = 51 × 50 − 51 = 2550 − 51 = 2499.

Answer: Answer: 51 × 49 = 2499

In plain English

Picture a big square made of tiles. Cut a smaller square of tiles out of one corner. What is left looks like the letter L. The number of tiles left is the big square minus the small square.

Now here is the surprise. You can cut that L into two rectangles and slide them together into one new rectangle. The sides of the new rectangle are the big side plus the small side, and the big side minus the small side.

That is the whole idea: a2 − b2 = (a + b)(a − b). It only works with a minus sign between two squares. A plus sign, like in x2 + 9, does not split this way.

Two habits help. First, look for a common factor before anything else. Second, check your answer by multiplying it back out. The middle terms should cancel.

See it

Cut the L and slide: a2 − b2 becomes (a + b)(a − b) b2 a2 − b2 b a a 1. Start with an L-shape A big square (side a) with a small square (side b) cut out of one corner. Its area is a2 − b2. B A a − b b a − b a 2. Cut it into two rectangles A is a wide and a − b tall. B is a − b wide and b tall. B sits above A, at the left. A B a + b a − b a b 3. Turn B on its side and slide it to the end of A. Now B is b wide and a − b tall. Together: one rectangle, a + b wide and a − b tall. Its area is (a + b)(a − b). a2 − b2 = (a + b)(a − b) Drawn with a = 5 and b = 2: 25 − 4 = 21, and 7 × 3 = 21. The picture needs a > b > 0. The algebra works for every number.
Cut the L into two rectangles, turn one on its side and slide it over: the same area is now a rectangle (a + b) wide and (a − b) tall, so a2 − b2 = (a + b)(a − b).
Factor x4 − 16 in layers x4 − 16 = (x2)2 − 42 so a = x2 and b = 4 (a + b) (a − b) x2 + 4 A sum of squares: prime, so stop here. x2 − 4 = x2 − 22, a difference of squares: split again. x + 2 x − 2 prime prime x4 − 16 = (x2 + 4)(x + 2)(x − 2) Completely factored: every factor is prime. Check: (x + 2)(x − 2) = x2 − 4, and (x2 + 4)(x2 − 4) = x4 − 16.
Factor x4 − 16 in layers. A difference of squares splits again, and a sum of squares is prime, so the branch stops.

The same idea, the fun way

Pick the one that clicks. Each one explains the same concept from a different angle.

Relationships

Settling up

Last week you paid 8 dollars for your friend’s lunch. This week your friend paid 8 dollars for yours. Now you are even: the amount you are owed and the amount you owe are the same size, in opposite directions, so they cancel. If your friend had paid only 5 dollars, you would not be even.

How it connects: When you multiply (a + b)(a − b), the middle terms are −ab and +ab. Same size, opposite signs, so they cancel and only a2 − b2 is left. The limit: cancelling needs equal sizes and opposite signs. In (a + b)(a + b) the middle terms have the same sign and add to 2ab. In (x + 3)(x − 2) they are −2x and +3x, which are different sizes, so +x is left over.

Movie

Dreams inside dreams

In the movie Inception, a team enters a person’s dream, and some missions go into a dream inside that dream, and then deeper still. To finish the job they go down one layer at a time.

How it connects: Factoring x4 − 16 works like that. The first layer is (x2 + 4)(x2 − 4). The factor x2 − 4 is itself a difference of squares, so you go one layer deeper: (x + 2)(x − 2). The factor x2 + 4 is a sum of squares, so it is the bottom layer and you stop. The limit: there is not always a deeper layer. x2 + 9 has nothing below it, and x2 − 7 has nothing below it either. Unlike a dream, you can always check by multiplying back up.

Proverb

Don’t judge a book by its cover

This proverb means you should not decide what something is really like from its outward appearance alone. The expression 27x2 − 12 looks like it cannot be a difference of squares, because neither 27 nor 12 is a perfect square.

How it connects: Look inside by taking out the GCF: 27x2 − 12 = 3(9x2 − 4) = 3(3x + 2)(3x − 2). The limit: the proverb says to look deeper, not that every cover hides a surprise. 2x2 − 5 has no GCF to take out and is still not a difference of squares, and x2 + 9 really is as plain as it looks.

Idiom

Hidden in plain sight

Something “hidden in plain sight” is right in front of you, but easy to miss because you are not looking for it. The expression m10 − 9 does not look like a difference of squares at first glance.

How it connects: But m10 = (m5)2 and 9 = 32, so m10 − 9 = (m5)2 − 32 = (m5 + 3)(m5 − 3). The squares are there if you halve the exponent. The limit: only an even exponent hides a square. m9 − 9 is not a difference of squares, because no whole-number power of m, squared, gives m9.

Food

The brownie pan

A square pan of brownies is 7 pieces by 7 pieces, so 49 pieces. Someone eats a 3-by-3 block of 9 pieces from one corner, and 40 pieces are left in an L-shape. Cut the L into a strip 7 pieces wide and 4 tall (28 pieces) and a block 4 wide and 3 tall (12 pieces). Turn the block on its side and slide it to the end of the strip: now you have a rectangle 10 pieces wide and 4 tall.

How it connects: Here a = 7 and b = 3: 72 − 32 = 49 − 9 = 40, and (7 + 3)(7 − 3) = 10 × 4 = 40. The limit: the picture needs the eaten corner to be smaller than the pan and the pieces to be whole. The algebra is true for any numbers, even ones that do not fit in a pan.

Gaming

The undo button

Many puzzle games have an undo button. Press it and you step back to the position before your last move, and redo takes you forward again. Multiplying (x + 5)(x − 5) gives x2 − 25. Factoring is the undo button: it takes x2 − 25 back to (x + 5)(x − 5).

How it connects: Checking a factorization is pressing redo: multiply the factors and see whether you land back on the start. The limit: an undo button always has an earlier move to go back to, but not every expression has factors. x2 + 25 was never made by multiplying two binomials with integer coefficients, so there is nothing to undo, and it is prime.

Everyday joke

The club with a rule

Why was the sum of squares turned away from the Difference of Squares Club? It couldn’t make a difference.

How it connects: The club’s rule is the minus sign: x2 − 9 = (x + 3)(x − 3), while x2 + 9 is prime. The limit: the joke is about the pattern, not about sums being worse. A sum of squares is a perfectly good factor, as in x4 − 16 = (x2 + 4)(x + 2)(x − 2).

Memory help

Acronym

FACTS

A checklist for factoring a difference of squares. F: take out any GCF First. A: Are there two terms, with one perfect square subtracted from another? C: Choose a and b, the things that were squared. T: write Two factors, (a + b)(a − b). S: Spot-check by expanding, and Scan the factors for anything that factors again. Say it as “FACTS.”

Rhyme

Root the first, root the last

Root the first, root the last; one plus, one minus, but you’re not done that fast. Expand to check: the middle is gone. If a middle term shows up, something went wrong. (This is for a minus sign between two perfect squares, a2 − b2. Take out any GCF first, and look again at each factor to see whether it factors more.)

Picture it

Picture the L

Picture a big square with a small square bitten out of one corner. That L-shaped piece is a2 − b2. Cut it in two and slide the pieces together into a rectangle that is a + b wide and a − b tall. If you can picture the rectangle, you can write the two factors.

Mnemonic

Root the number, halve the power

To find what was squared in a perfect-square term, take the square root of the number and cut each exponent in half. For 49x10: the root of 49 is 7 and half of 10 is 5, so it is (7x5)2. This only works when the number is a perfect square and the exponent is even.

Trap alert

A minus is not a square

x2 − 36 is not (x − 6)2. Expanding (x − 6)2 gives x2 − 12x + 36, which has a middle term. The factors of a difference of squares have opposite signs: (x + 6)(x − 6).

Trap alert

A plus sign stops the pattern

x2 + 64 is not (x + 8)(x − 8), because that product is x2 − 64. A two-term sum of squares like x2 + 64 is prime. Do not force it, and do not forget the coefficient: 25x2 − 64 is (5x + 8)(5x − 8), not (25x + 8)(25x − 8).

Study notes

Key ideas

  • A difference of squares is a binomial with one perfect square subtracted from another. It factors as a2 − b2 = (a + b)(a − b).
  • Factoring is the reverse of multiplying. Check by expanding: the middle terms −ab and +ab cancel.
  • A term is a perfect square if its coefficient is a perfect square and every variable exponent is even. Root the number and halve the exponents.
  • Take out any GCF first. Then keep factoring until the expression is completely factored: x4 − 16 = (x2 + 4)(x + 2)(x − 2).
  • After any GCF is taken out, a two-term sum of squares of degree 2, such as x2 + 9, is prime: it cannot be factored using integers, or even with real-number coefficients.
  • The pattern also gives mental math: (n + d)(n − d) = n2 − d2.

Steps

  1. F: take out any GCF first.
  2. A: confirm there are two terms, with one perfect square subtracted from another.
  3. C: choose a and b (root the numbers, halve the exponents).
  4. T: write two factors, (a + b)(a − b).
  5. S: spot-check by expanding, then scan each factor to see whether it factors again.

Formulas

  • Difference of squares: a2 − b2 = (a + b)(a − b)
  • Expanding the check: (a + b)(a − b) = a2 − ab + ab − b2 = a2 − b2
  • Mental math: (n + d)(n − d) = n2 − d2
  • A square of a power: (xk)2 = x2k, so halve the exponent to find the root

Watch out

  • A plus sign is not a difference. x2 + 9 cannot be factored using integers.
  • x2 − 9 = (x + 3)(x − 3), not (x − 3)2. The two factors have opposite signs.
  • Take the square root of the coefficient too: 25x2 − 64 = (5x + 8)(5x − 8).
  • Look for a GCF first. 18x2 − 50 has no perfect-square terms until the 2 comes out.
  • Stop only when no factor can be factored further. (x2 + 4)(x2 − 4) is not complete.
  • The pattern needs both terms to be perfect squares. x2 − 10 does not factor using integers.

Why it works

  • Why does the pattern work? Multiplying (a + b)(a − b) gives the middle terms −ab and +ab, which cancel, leaving a2 − b2.
  • Why does a two-term sum of squares like x2 + 9 not factor? Cancelling the middle term needs opposite signs, which makes the last term negative. And any factor (x + p) would make the product 0 at x = −p, but x2 + 9 is never 0 for real x.
  • Why does the mental-math trick work? A pair like 47 × 53 is a sum and a difference around 50, so it equals 502 − 32.

Common mistakes

✗ Mistakex2 − 25 = (x − 5)2 (using the same sign twice).
✓ Instead(x − 5)2 = x2 − 10x + 25 has a middle term. A difference of squares has opposite signs in its factors: x2 − 25 = (x + 5)(x − 5).
✗ Mistakex2 + 16 = (x + 4)(x − 4) (a sum of squares factors like a difference).
✓ Instead(x + 4)(x − 4) = x2 − 16. A two-term sum of squares such as x2 + 16 is prime: it cannot be factored using integers.
✗ Mistake4x2 − 81 = (4x + 9)(4x − 9) (the coefficient is not rooted).
✓ InsteadTake the square root of the coefficient too: 4x2 = (2x)2, so 4x2 − 81 = (2x + 9)(2x − 9). The wrong answer expands to 16x2 − 81.
✗ Mistakex4 − 16 = (x2 + 4)(x2 − 4) is completely factored.
✓ Insteadx2 − 4 is a difference of squares and factors again. The complete answer is (x2 + 4)(x + 2)(x − 2).
✗ Mistake18x2 − 50 cannot be factored, because 18 and 50 are not perfect squares.
✓ InsteadTake out the GCF first: 18x2 − 50 = 2(9x2 − 25) = 2(3x + 5)(3x − 5).

Teach it back

Explain to a friend how to factor 45x2 − 20 completely, why you look for a GCF first, and why x2 + 4 cannot be factored.

Explaining it out loud shows what you really know. Did you…

  • I took out the GCF 5 first, getting 5(9x2 − 4), because 45 and 20 are not perfect squares.
  • I found a = 3x and b = 2, wrote 5(3x + 2)(3x − 2), and checked by expanding: (3x + 2)(3x − 2) = 9x2 − 4 and 5(9x2 − 4) = 45x2 − 20.
  • I explained that a sum of squares like x2 + 4 is prime: opposite signs would make a minus sign, and equal signs would make a middle term.

Your 3-week memory plan

  1. TodayRead the lesson, then try the flashcards.
  2. TomorrowRecall it from memory first, then check.
  3. In 3 daysTake the true/false and practice questions.
  4. In 1 weekMix it with older topics and teach it back.
  5. In 3 weeksA last quick review before it fades.

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Special Products of BinomialsFactoring Out the Greatest Common FactorFactoring TrinomialsFactoring by GroupingMultiplying PolynomialsSolving Quadratics by Factoring