Equivalent Quadratic Expressions

Jump to section:

Key Ideas Easy Medium Hard Super Challenge

What to Remember

Same polynomial, different form

Factored form, standard form, and an area picture can all be the same quadratic.

Expand to check

Multiply the factors. If you get the same standard form, they are equal.

A graph check

Equal expressions have the same graph. Same zeros, same y-intercept.

Tiles

A rectangle of tiles (x + 3) by (x + 2) is the same collection as x² + 5x + 6 unit pieces.

Level 1

Easy

Expand a factored form and match.

Example

Show that (x + 2)(x + 5) = x² + 7x + 10.

x² + 5x + 2x + 10

x² + 7x + 10

Answer: x² + 7x + 10

Show that (x − 3)(x + 3) = x² − 9.

Show Answer

x² + 3x − 3x − 9

x² − 9

Answer: x² − 9

Is (x + 1)² equal to x² + 1? Expand.

Show Answer

x² + 2x + 1

Not equal to x² + 1.

Answer: no

Match x² + 6x + 9 to a square of a binomial.

Show Answer

(x + 3)²

Answer: (x + 3)²

A tile picture is an x² square, five x-tiles, and six units. Write two equivalent expressions.

Show Answer

x² + 5x + 6 and (x + 2)(x + 3)

Answer: x² + 5x + 6 = (x + 2)(x + 3)

Show 2(x + 4) = 2x + 8.

Show Answer

Distribute: 2x + 8.

Answer: 2x + 8

Level 2

Medium

a > 1, or a graph/zeros check.

Example

Show that (2x + 1)(x + 3) = 2x² + 7x + 3.

2x² + 6x + x + 3

2x² + 7x + 3

Answer: 2x² + 7x + 3

The graph of y = (x − 1)(x − 4) has zeros 1 and 4. Expand and find the y-intercept.

Show Answer

y = x² − 5x + 4

y-intercept 4, which is (−1)(−4).

Answer: x² − 5x + 4; (0, 4)

Is 3(x² + 2x + 1) equal to 3(x + 1)²?

Show Answer

x² + 2x + 1 = (x + 1)², so yes.

Answer: yes

Show x(x + 6) + 9 = (x + 3)².

Show Answer

x² + 6x + 9

(x + 3)² = x² + 6x + 9

Answer: (x + 3)²

Two area models: (x + 1) by (x + 5) and (x + 2) by (x + 4). Same area?

Show Answer

x² + 6x + 5 vs x² + 6x + 8. No. Same linear term, different constant.

Answer: no

Write x² − 4x + 4 in three equivalent forms: standard, factored, and a square.

Show Answer

x² − 4x + 4

(x − 2)(x − 2)

(x − 2)²

Answer: (x − 2)²

Level 3

Hard

Justify with more than one representation.

Example

Show 2x² + 10x + 12 = 2(x + 2)(x + 3) with tiles in words, with symbols, and by comparing graphs of y = 2x² + 10x + 12 and y = 2(x + 2)(x + 3).

Symbols: 2(x² + 5x + 6) = 2(x + 2)(x + 3).

Tiles: two copies of an (x + 2) by (x + 3) rectangle.

Graphs: both open up, zeros at −2 and −3, y-intercept 12.

Answer: 2(x + 2)(x + 3)

Find k so that x² + kx + 9 = (x + 3)².

Show Answer

(x + 3)² = x² + 6x + 9

k = 6

Answer: k = 6

A student says (2x + 6)(x + 1) and 2(x + 3)(x + 1) look different so they are not equal. Expand both.

Show Answer

(2x + 6)(x + 1) = 2x² + 2x + 6x + 6 = 2x² + 8x + 6

2(x + 3)(x + 1) = 2(x² + 4x + 3) = 2x² + 8x + 6

Equal. The first has a GCF still sitting in one binomial.

Answer: equal

Show x² − 1 and (x − 1)(x + 1) have the same zeros and the same y-intercept.

Show Answer

Zeros ±1

y-intercept −1 both ways.

Answer: zeros ±1; intercept −1

Write two different-looking but equal expressions for the area of a square of side x + 4.

Show Answer

(x + 4)² and x² + 8x + 16

Answer: (x + 4)² = x² + 8x + 16

Is x² + 5x + 6 equal to (x + 1)(x + 6)? Check two ways.

Show Answer

Expand: x² + 7x + 6, not 5x.

Zeros of (x + 1)(x + 6) are −1 and −6. Zeros of x² + 5x + 6 are −2 and −3.

Answer: no

Super Challenge

Prove (x + a)(x + b) = x² + (a + b)x + ab by distributing, and name the properties.

Show Answer

x(x + b) + a(x + b)

Distributive property.

x² + bx + ax + ab

x² + (a + b)x + ab

Answer: x² + (a + b)x + ab

A graph of y = x² + 4x + 4 looks like a parabola touching the x-axis once. Write two equivalent expressions that make that obvious.

Show Answer

(x + 2)²

(x + 2)(x + 2)

Answer: (x + 2)²

Give a verbal, a symbolic, and a tile reason that x(x + 2) + 3(x + 2) equals (x + 3)(x + 2).

Show Answer

Verbal: x groups of (x + 2) plus 3 groups of (x + 2) is (x + 3) groups.

Symbolic: factor (x + 2).

Tiles: a rectangle (x + 3) by (x + 2).

Answer: (x + 3)(x + 2)

Need help showing expressions are equal?

Book a Session with Amy