Factoring Quadratic Equations
As mentioned in the previous algebra lesson, when factoring Quadratic Equations, of the form:
ax^{2} + bx + c = 0 where a, b and c are numbers and a ≠ 0.
we try to find common factors, and then look for patterns that will help you
to factorize the quadratic equation. For example: Square of Sum,
Square of Difference and Difference of
Two Squares.
In other cases, you will have to try out different possibilities to get the
right factors for quadratic equations.
To factorize quadratic equations of the form: x^{2} + 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: x^{2} + 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) = x^{2} + 5x + 2x + 10 = x^{2} + 7x + 10
The factors are (x + 2) (x + 5)
Step 4: Going back to the original quadratic equation
x^{2} + 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: x^{2} + 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)= x^{2} + 5x – x – 5 = x^{2} + 4x – 5
Step 4: Going back to the original quadratic equation
x^{2} + 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: x^{2} – 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) = x^{2} – 6 x + x – 6 = x^{2} – 5x – 6
Step 4: Going back to the original quadratic equation
x^{2} – 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: x^{2} – 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) = x^{2} – 4 x – 2x + 8 = x^{2} – 6x + 8
Step 4: Going back to the original quadratic equation
x^{2} – 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
Sometimes the coefficient of x in quadratic equations may not be 1 but the expression can be simplified by finding common factors.
When the coefficient of x^{2} is greater than 1 and we cannot simplify the quadratic equation by finding common factors, we would need to consider the factors of the coefficient of x^{2} and the factors of c in order to get the numbers whose sum is b. If there are many factors to consider you may want to use the quadratic formula instead.
Example 1: Get the values of x for the equation 2x^{2} – 14x + 20 = 0
Step 1: Find common factors if you can.
2x^{2} – 14x + 20 = 2(x^{2} – 7x + 10)
Step 2: Find the factors of (x^{2} – 7x + 10)
List out the factors of 10:
We need to get the negative factors of 10 to get a negative sum.
–1 × –10, –2 × –5
Step 3: Find the factors whose sum is – 7:
1 + ( –10) ≠ –7
–2 + ( –5) = –7
Step 4: Write out the factors and check using the distributive property.
2(x – 2) (x – 5) = 2(x^{2} – 5 x – 2x + 10)
= 2(x^{2} – 7x + 10) = 2x^{2} – 14x + 20
Step 5: Going back to the original equation
2x^{2} – 14x + 20 = 0 Factorize the left hand side of the equation
2(x – 2) (x – 5) = 0
We get two values for x
Answer: x = 2, x = 5
Example 2: Get the values of x for the equation 7x^{2} + 18x + 11 = 0
Step 1: List out the factors of 7 and 11
Factors of 7:
1 × 7
Factors of 11:
1 × 11
Since 7 and 11 are prime numbers there are only two possibilities to try out.
Step 2: Write down the different combinations of the factors and perform the distributive property to check.
(7x + 1)(x + 11) ≠ 7x^{2} + 18x + 11
(7x + 11)(x + 1) = 7x^{2} + 18x + 11
Step 3: Write out the factors and check using the distributive property.
(7x + 11)(x + 1) = 7x^{2} + 7x + 11x + 11 = 7x^{2} + 18x + 11
Step 4: Going back to the original equation
7x^{2} + 18x + 11= 0 Factorize the left hand side of the equation
(7x + 11)(x + 1) = 0
We get two values for x
Answer:
Example 3: Get the values of x for the equation 4x^{2} + 26x + 12 = 0
Step 1: List out the factors of 4 & 12
Factors of 4:
1 × 4, 2 × 2
Factors of 12:
1 × 12, 2 × 6, 3 × 4
Step 2: Write down the different combinations of the factors and perform the distributive property to check. When there are many factors to check, this becomes a tedious method to solve such quadratic equations, so you may want to try the quadratic formula instead.
(4x + 12)(x + 1) ≠ 4x^{2} + 26x + 12
(4x + 12)(x + 12) ≠ 4x^{2} + 26x + 12
(4x + 2)(x + 6) ≠ 4x^{2} + 26x + 12
(4x + 6)(x + 2) ≠ 4x^{2} + 26x + 12
(4x + 3)(x + 4) ≠ 4x^{2} + 26x + 12
(4x + 4)(x + 3) ≠ 4x^{2} + 26x + 12
(2x + 12)(2x + 1) = 4x^{2} + 26x + 12
(2x + 2)(2x + 6) ≠ 4x^{2} + 26x + 12
(2x + 3)(2x + 4) ≠ 4x^{2} + 26x + 12
Step 3: Going back to the original quadratic equation
4x^{2} + 26x + 12 = 0 Factorize the left side of the equation
(2x + 12)(2x + 1) = 0
We get two values for x
Answer:
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