Solving quadratics by factoring worksheet answer key

Problem 1 :

Solve the quadratic equation by factoring :

x2 – 5x – 24  =  0

Problem 2 :

Solve the quadratic equation by factoring :

3x2 – 5x – 12  =  0

Problem 3 :

Solve the quadratic equation by factoring :

(x + 4)2  =  2x + 88

Problem 4 :

Solve the quadratic equation by factoring :

√5x2 + 2x – 3√5  =  0

Problem 5 :

Solve the quadratic equation by factoring :

x - 18/x  =  3

Detailed Answer Key

Problem 1 :

Solve the quadratic equation by factoring :

x2 – 5x – 24  =  0

Solution :

In the given quadratic equation, the coefficient of x2 is 1.

Decompose the constant term -24 into two factors such that the product of the two factors is equal to -24 and the addition of two factors is equal to the coefficient of x, that is 5. 

Then, the two factors of -24 are 

+3 and -8

Factor the given quadratic equation using +3 and -8 and solve for x.

(x + 3)(x - 8)  =  0

x + 3  =  0  or  x - 8  =  0

x  =  -3  or  x  =  8

So, the solution is {-3, 8}. 

Problem 2 :

Solve the quadratic equation by factoring :

3x2 – 5x – 12  =  0

Solution :

In the given quadratic equation, the coefficient of x2 is not 1.

So, multiply the coefficient of x2 and the constant term "-12". 

3 ⋅ (-12)  =  -36

Decompose -36 into two factors such that the product of two factors is equal to -36 and the addition of two factors is equal to the coefficient of x, that is -5.

Then, the two factors of -36 are 

+4 and -9

Now we have to divide the two factors 4 and -9 by the coefficient of x2, that is 3.

Now, factor the given quadratic equation and solve for x as shown below. 

(3x + 4)(x - 3)  =  0

3x + 4  =  0  or  x - 3  =  0

x  =  -4/3  or  x  =  3

So, the solution is {-4/3, 3}. 

Problem 3 :

Solve the quadratic equation by factoring :

(x + 4)2  =  2x + 88

Solution :

Write the given quadratic equation in the form

ax2 + bx + c  =  0

Then, 

(x + 4)2  =  2x + 88

(x + 4)(x + 4)  =  2x + 88

x2 + 4x + 4x + 16  =  2x + 88

x2 + 8x + 16  =  2x + 88

x2 + 6x - 72  =  0

In the quadratic equation above, the coefficient of x2 is 1.

Decompose the constant term -72 into two factors such that the product of the two factors is equal to -72 and the addition of two factors is equal to the coefficient of x, that is +6. 

Then, the two factors of -72 are 

+12 and -6

Factor the given quadratic equation using +12 and -6 and solve for x.

(x + 12)(x - 6)  =  0

x + 12  =  0  or  x - 6  =  0

x  =  -12  or  x  =  6

So, the solution is {-12, 6}. 

Problem 4 :

Solve the quadratic equation by factoring :

√5x2 + 2x – 3√5  =  0

Solution :

In the given quadratic equation, the coefficient of x2 is not 1.

So, multiply the coefficient of x2 and the constant term "-3√5". 

√5 ⋅ (-3√5)  =  -15

Decompose -15 into two factors such that the product of two factors is equal to -15 and the addition of two factors is equal to the coefficient of x, that is +2.

Then, the two factors of -15 are 

+5 and -3

Now we have to divide the two factors +5 and -3 by the coefficient of x2, that is √5.

Now, factor the given quadratic equation and solve for x as shown below. 

(x + √5)(√5x - 3)  =  0

x + √5  =  0  or  √5x - 3  =  0

x  =  -√5  or  x  =  3/√5

So, the solution is {-√5, 3/√5}. 

Problem 5 :

Solve the quadratic equation by factoring :

x - 18/x  =  3

Solution :

Write the given quadratic equation in the form

ax2 + bx + c  =  0

Then, 

x - 18/x  =  3

x2/x - 18/x  =  3

(x2 - 18)/x  =  3

x2 - 18  =  3x

x2 - 3x - 18  =  0

In the quadratic equation above, the coefficient of x2 is 1.

Decompose the constant term -18 into two factors such that the product of the two factors is equal to -18 and the addition of two factors is equal to the coefficient of x, that is -3. 

Then, the two factors of -18 are 

+3 and -6

Factor the given quadratic equation using +3 and -6 and solve for x.

(x + 3)(x - 6)  =  0

x + 3  =  0  or  x - 6  =  0

x  =  -3  or  x  =  6

So, the solution is {-3, 6}. 

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