Factoring Quadratic Equations - 2

In some cases, you will have to try out different possibilities to get the right factors for quadratic equations.

If the Coefficient Of x² Is 1

To factorize quadratic equations of the form: x² + bx + c, you will need to find two numbers whose product is c and whose sum is b.

Example 1: (b and c are both positive)

Solve the quadratic equation: x² + 7x + 10 = 0

Step 1: List out the factors of 10:

1 × 10, 2 × 5

Step 2: Find the factors whose sum is 7:

1 + 10 ≠ 7
2 + 5 = 7  

Step 3: Write out the factors and check using the distributive property.

(x + 2) (x + 5) = x² + 5x + 2x + 10 = x² + 7x + 10
The factors are (x + 2) (x + 5)

Step 4: Going back to the original quadratic equation

x² + 7x + 10 = 0       Factorize the left side of the quadratic equation
(x + 2) (x + 5) = 0

We get two values for x.

Answer:   x = – 2, x = – 5

Example 2: (b is positive and c is negative)

Get the values of x for the equation: x² + 4x – 5 = 0

Step 1: List out the factors of – 5:

1 × –5, –1 × 5

Step 2: Find the factors whose sum is 4:

1 – 5 ≠ 4
–1 + 5 = 4  

Step 3: Write out the factors and check using the distributive property.

(x – 1)(x + 5)= x² + 5x – x – 5 = x² + 4x – 5

Step 4: Going back to the original quadratic equation

x² + 4x – 5 = 0       Factorize the left hand side of the equation
(x – 1)(x + 5) = 0

We get two values for x.

Answer:   x = 1, x = – 5

Example 3: (b and c are both negative)

Get the values of x for the equation: x² – 5x – 6

Step 1: List out the factors of – 6:

1 × –6, –1 × 6, 2 × –3, –2 × 3

Step 2: Find the factors whose sum is –5:

1 + ( –6) = –5  

Step 3: Write out the factors and check using the distributive property.

(x + 1) (x – 6) = x² – 6 x + x – 6 = x² – 5x – 6

Step 4: Going back to the original quadratic equation

x² – 5x – 6 = 0       Factorize the left hand side of the equation
(x + 1) (x – 6) = 0

We get two values for x.

Answer: x = –1, x = 6

Example 4: (b is negative and c is positive)

Get the values of x for the equation: x² – 6x + 8 = 0

Step 1: List out the factors of 8:

We need to get the negative factors of 8 to get a negative sum.
–1 × – 8, –2 × –4

Step 2: Find the factors whose sum is – 6:

–1 + ( –8) ≠ –6
–2 + ( –4) = –6  

Step 3: Write out the factors and check using the distributive property.

(x – 2) (x – 4) = x² – 4 x – 2x + 8 = x² – 6x + 8

Step 4: Going back to the original quadratic equation

x² – 6x + 8 = 0       Factorize the left hand side of the equation
(x – 2) (x – 4) = 0

We get two values for x.

Answer: x = 2, x = 4

The following video will show more examples of solving quadratic equations.

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