SOLVING QUADRATIC EQUATIONS BY FACTORING WORKSHEET

Solve the following quadratic equations by factoring.

Problem 1 :

x² + 7x + 12 = 0

Problem 2 :

x² - 9x + 20 = 0

Problem 3 :

x² – 5x – 24 = 0

Problem 4 :

3x² – 5x – 12 = 0

Problem 5 :

(x + 4)² = 2x + 88

Problem 6 :

√5x² + 2x – 3√5 = 0

Problem 7 :

x - ¹⁸⁄ₓ = 3

Problem 8 :

x² - 100 = 0

Problem 9 :

9x² - 64 = 0

Problem 10 :

(3x + 2)² - 81 = 0

Answers

1. Answer :

In the quadratic equation x² + 5x + 6 = 0, multiply the coefficient of x² is 1 and the consterm 12.

= 1 x 12

= 12

Find two factors of 12 such that the product of them is equal to 12 and the sum is equal to the coeffient of x, 7.

3 x 4 = 12

3 + 4 = 7

So, the two factors of 12 are 3 and 4.

Split the middle term 7using the two factors 3 and 4.

x² + 3x + 4x + 12 = 0

Factor and solve the above quadratic equation by grouping.

(x² + 3x) + (4x + 12) = 0

x(x + 3) + 4(x + 3) = 0

(x + 3)(x + 4) = 0

x + 3 = 0  or  x + 4 = 0

x = -3  or  x = -4

2. Answer :

In the quadratic expression x² - 9x + 20 = 0, multiply the coefficient of x² is 1 and the consterm 20.

= 1 x 20

= 20

Find two factors of 20 such that the product of them is equal to 20 and the sum is equal to the coeffient of x, -9.

(-4) x (-5) = 20

(-4) + (-5) = -9

So, the two factors of 20 are -4 and -5.

Split the middle term 10x using the two factors -4 and -5.

x² - 4x - 5x + 20 = 0

Factor and solve the above quadratic equation by grouping.

(x² - 4x) + (-5x + 20) = 0

x(x - 4) - 5(x - 4) = 0

(x - 4)(x - 5) = 0

x - 4 = 0  or  x - 5 = 0

x = 4  or  x = 5

3. Answer :

In the quadratic expression x² – 5x – 24 = 0, multiply the coefficient of x², 1 and the consterm -24.

= 1 x (-24)

= -24

Find two factors of -24 such that the product of them is equal to -24 and the sum is equal to the coeffient of x, -5.

-8 x 3 = -24

-8 + 3 = -5

So, the two factors of -24 are -8 and 3.

Split the middle term -5x using the two factors -8 and 3.

x² - 8x + 3x - 24 = 0

Factor and solve the above quadratic equation by grouping.

(x² - 8x) + (3x - 24) = 0

x(x - 8) + 3(x - 8) = 0

(x - 8)(x + 3) = 0

x - 8 = 0  or  x + 3 = 0

x = 8  or  x = -3

4. Answer :

In the quadratic expression 3x² – 5x – 12 = 0, multiply the coefficient of x², 3 and the consterm -12.

= 3 x (-12)

= -36

Find two factors of -36 such that the product of them is equal to -36 and the sum is equal to the coeffient of x, -5.

-9 x 4 = -36

-9 + 4 = -5

So, the two factors of -36 are -9 and 4.

Split the middle term -5x using the two factors -9 and 4.

3x² - 9x + 4x - 36 = 0

Factor and solve the above quadratic equation by grouping.

(3x² - 9x) + (4x - 36) = 0

3x(x - 9) + 4(x - 9) = 0

(x - 9)(3x + 4) = 0

x - 9 = 0  or  3x + 4 = 0

x = 9  or  x = -⁴⁄₃

5. Answer :

The quadratic expression (x + 4)² = 2x + 88 is not in standard form. Write it in standard form.

(x + 4)² = 2x + 88

(x + 4)(x + 4) = 2x + 88

x² + 4x + 4x + 16 = 2x + 88

x² + 8x + 16 = 2x + 88

Subtract 2x and 88 from both sides.

x² + 6x - 72 = 0

In the quadratic equation x² + 6x - 72 = 0, multiply the coefficient of x², 1 and the constant term -72.

= 1 x (-72)

= -72

Find two factors of -72 such that the product of them is equal to -72 and the sum is equal to the coeffient of x, 6.

-6 x 12 = -72

-6 + 12 = 6

So, the two factors of -72 are -6 and 12.

Split the middle term 6x using the two factors -6 and 12.

x² - 6x + 12x - 72 = 0

Factor and solve the above quadratic equation by grouping.

(x² - 6x) + (12x - 72) = 0

x(x - 6) + 12(x - 6) = 0

(x - 6)(x + 12) = 0

x - 6 = 0  or  x + 12 = 0

x = 6  or  x = -12

6. Answer :

In the quadratic expression √5x² + 2x – 3√5 = 0, multiply the coefficient of x², √5 and the consterm -3√5.

= √5 x (-3√5)

= -15

Find two factors of -15 such that the product of them is equal to -15 and the sum is equal to the coeffient of x, 2.

-3 x 5 = -15

-3 + 5 = 2

So, the two factors of -15 are -3 and 5.

Split the middle term 2x using the two factors -3 and 5.

√5x² - 3x + 5x – 3√5 = 0

Factor and solve the above quadratic equation by grouping.

(√5x² - 3x) + (5x – 3√5) = 0

(√5x² - 3) + (√5√5x - 3√5) = 0

x(√5x - 3) + √5(√5x - 3) = 0

(√5x - 3)(x + √5) = 0

x - 3 = 0  or  x + √5 = 0

x = ³⁄√₅  or  x = -√5

7. Answer :

The quadratic expression x - ¹⁸⁄ₓ = 3 is not in standard form. Write it in standard form.

x - ¹⁸⁄ₓ = 3

Multiply both sides by x.

x² - 18 = 3x

Subtract 3x from both sides.

x² - 3x - 18 = 0

In the quadratic expression x² - 3x - 18 = 0, multiply the coefficient of x², 1 and the consterm -18.

= 1 x (-18)

= -18

Find two factors of -18 such that the product of them is equal to -18 and the sum is equal to the coeffient of x, -3.

-6 x 3 = -18

-6 + 3 = -3

So, the two factors of -18 are -6 and 3.

Split the middle term -3x using the two factors -6 and 3.

x² - 6x + 3x - 18 = 0

Factor and solve the above quadratic equation by grouping.

(x² - 6x) + (3x - 18) = 0

x(x - 6) + 3(x - 6) = 0

(x - 6)(x + 3) = 0

x - 6 = 0  or  x + 3 = 0

x = 6  or  x = -3

8. Answer :

The quadratic equation x² - 100 = 0 can be solved using the following algebraic identity.

a² - b² = (a + b)(a - b)

x² - 100 = 0

x² - 10² = 0

(x + 10)(x - 10) = 0

x + 10 = 0  or  x - 10 = 0

x = -10  or  x = 10

9. Answer :

9x² - 64 = 0

3²x² – 8² = 0

(3x)² – 8² = 0

(3x + 8)(3x - 8) = 0

3x + 8 = 0  or  3x - 8 = 0

x = -⁸⁄₃  or  x = ⁸⁄₃

10. Answer :

(3x + 2)² - 81 = 0

(3x + 2)² – 9² = 0

(3x + 2 + 9)(3x + 2 - 9) = 0

(3x + 11)(3x - 7) = 0

3x + 11 = 0  or  3x - 7 = 0

x = -¹¹⁄₃  or  x = ⁷⁄₃

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