To solve a quadratic equation by factoring, the quadratic equation has ro be in standard form, that is
ax^{2} + bx + c = 0
In the above standard form of a quadratic equation,
coefficient of x^{2} = a
coefficient of x = b
constant term = c
The following steps can be used to solve a quadratic equation by factoring.
Step 1 :
Multiply the coefficient of x^{2}, a and the constant term c.
= ac
Step 2 :
Resolve ac into two factors, say m and n such that the product of them is equal to ac and the sum is equal to the coefficient of x, b.
mn = ac
m + n = b
Step 3 :
Split the middle term bx using the two numbers m and n.
ax^{2} + mx + nx + c = 0
Step 4 :
Factor and solve the quadratic equation in step 3 by grouping as shown below.
(ax^{2} + mx) + (nx + c) = 0
Solve each of the following quadratic equations by factoring.
Example 1 :
x^{2} + 5x + 6 = 0
Solution :
Step 1 :
In the quadratic equation x^{2} + 5x + 6 = 0, multiply the coefficient of x^{2}, 1 and the constant term 6.
= 1 x 6
= 6
Step 2 :
Resolve 6 into two factors such that the product of them is equal to 6 and the sum is equal to the coeffient of x, 5.
2 x 3 = 6
2 + 3 = 5
So, the two factors of 6 are 2 and 3.
Step 3 :
Split the middle term 5x using the two factors 2 and 3.
x^{2} + 2x + 3x + 6 = 0
Step 4 :
Factor and solve the quadratic equation in the above step 3 by grouping.
(x^{2} + 2x) + (3x + 6) = 0
x(x + 2) + 3(x + 2) = 0
(x + 2)(x + 3) = 0
x + 2 = 0 or x + 3 = 0
x = -2 or x = -3
Example 2 :
x^{2} + 10x + 24 = 0
Solution :
Step 1 :
In the quadratic equation x^{2} + 10x + 24 = 0, multiply the coefficient of x^{2}, 1 and the constant term 24.
= 1 x 24
= 24
Step 2 :
Resolve 24 into two factors such that the product of them is equal to 24 and the sum is equal to the coeffient of x, 10.
4 x 6 = 24
4 + 6 = 10
So, the two factors of 24 are 4 and 6.
Step 3 :
Split the middle term 10x using the two factors 4 and 6.
x^{2} + 4x + 6x + 24 = 0
Step 4 :
Factor and solve the quadratic equation in the above step 3 by grouping.
(x^{2} + 4x) + (6x + 24) = 0
x(x + 4) + 6(x + 4) = 0
(x + 4)(x + 6) = 0
x + 4 = 0 or x + 6 = 0
x = -4 or x = -6
Example 3 :
x^{2} - 7x + 10 = 0
Solution :
Step 1 :
In the quadratic equation x^{2} - 7x + 10 = 0, multiply the coefficient of x^{2}, 1 and the constant term 10.
= 1 x 10
= 10
Step 2 :
Resolve 10 into two factors such that the product of them is equal to 10 and the sum is equal to the coeffient of x, -7.
(-5) x (-2) = 10
(-5) + (-2) = -7
So, the two factors of 10 are -5 and -2.
Step 3 :
Split the middle term -7x using the two factors -5 and -2.
x^{2} - 5x - 2x + 10 = 0
Step 4 :
Factor and solve the quadratic equation in the above step 3 by grouping.
(x^{2} - 5x) + (-2x + 10) = 0
x(x - 5) - 2(x - 5) = 0
(x - 5)(x - 2) = 0
x - 5 = 0 or x - 2 = 0
x = 5 or x = 2
Example 4 :
x^{2} + 4x - 12 = 0
Solution :
Step 1 :
In the quadratic equation x^{2} + 4x - 12 = 0, multiply the coefficient of x^{2}, 1 and the constant term -12.
= 1 x (-12)
= -12
Step 2 :
Resolve -12 into two factors such that the product of them is equal to -12 and the sum is equal to the coeffient of x, 4.
-2 x 6 = -12
-2 + 6 = 4
So, the two factors of -12 are -2 and 6.
Step 3 :
Split the middle term 4x using the two factors -2 and 6.
x^{2} - 2x + 6x - 12 = 0
Step 4 :
Factor and solve the quadratic equation in the above step 3 by grouping.
(x^{2} - 2x) + (6x - 12) = 0
x(x - 2) + 6(x - 2) = 0
(x - 2)(x + 6) = 0
x - 2 = 0 or x + 6 = 0
x = 2 or x = -6
Example 5 :
x^{2} - x - 6 = 0
Solution :
Step 1 :
In the quadratic equation x^{2} - x - 6 = 0, multiply the coefficient of x^{2}, 1 and the constant term -6.
= 1 x (-6)
= -6
Step 2 :
Resolve -6 into two factors such that the product of them is equal to -6 and the sum is equal to the coeffient of x, -1.
-3 x 2 = -6
-3 + 2 = -1
So, the two factors of -6 are -3 and 2.
Step 3 :
Split the middle term -x using the two factors -3 and 2.
x^{2} - 3x + 2x - 6 = 0
Step 4 :
Factor and solve the quadratic equation in the above step 3 by grouping.
(x^{2} - 3x) + (2x - 6) = 0
x(x - 3) + 2(x - 3) = 0
(x - 3)(x + 2) = 0
x - 3 = 0 or x + 2 = 0
x = 3 or x = -2
Example 6 :
2x^{2} + 7x + 6 = 0
Solution :
Step 1 :
In the quadratic equation 2x^{2} + 7x + 6 = 0, multiply the coefficient of x^{2}, 2 and the constant term 6.
= 2 x 6
= 12
Step 2 :
Resolve 12 into two factors such that the product of them is equal to 12 and the sum is equal to the coeffient of x, 8.
3 x 4 = 12
3 + 4 = 7
So, the two factors of 12 are 3 and 4.
Step 3 :
Split the middle term 7x using the two factors 3 and 4.
2x^{2} + 3x + 4x + 6 = 0
Step 4 :
Factor and solve the quadratic equation in the above step 3 by grouping.
(2x^{2} + 3x) + (4x + 6) = 0
x(2x + 3) + 2(2x + 3) = 0
(2x + 3)(x + 2) = 0
2x + 3 = 0 or x + 2 = 0
x = ⁻³⁄₂ or x = -2
Example 7 :
-2x^{2} - 2x + 5 = -5x^{2} + 13
Solution :
Step 1 :
The given quadratic equation is not in standard form. Write it in standard form.
-2x^{2} - 2x + 5 = -5x^{2} + 13
Add 5x^{2 }to both sides.
3x^{2} - 2x + 5 = 13
Subtract 13 from both sides.
3x^{2} - 2x - 8 = 0
Step 2 :
In the quadratic expression 3x^{2} - 2x - 8 = 0, multiply the coefficient of x^{2}, 3 and the constant term -8.
= 3 x (-8)
= -24
Step 3 :
Resolve -24 into two factors such that the product of them is equal to -24 and the sum is equal to the coeffient of x, -2.
-6 x 4 = -24
-6 + 4 = -2
So, the two factors of 12 are -6 and 4.
Step 3 :
Split the middle term -2x using the two factors -6 and 4.
3x^{2} - 6x + 4x - 8 = 0
Step 4 :
Factor and solve the quadratic equation in the above step 3 by grouping.
(3x^{2} - 6x) + (4x - 8) = 0
3x(x - 2) + 4(x - 2) = 0
(x - 2)(3x + 4) = 0
x - 2 = 0 or 3x + 4 = 0
x = 2 or x = ⁻⁴⁄₃
Example 8 :
x^{2} - 25 = 0
Solution :
The quadratic equation x^{2} - 25 = 0 can be solved using the following algebraic identity.
a^{2} - b^{2} = (a + b)(a - b)
x^{2} - 25 = 0
x^{2} - 5^{2} = 0
(x + 5)(x - 5) = 0
x + 5 = 0 or x - 5 = 0
x = -5 or x = 5
Example 9 :
4x^{2} - 49 = 0
Solution :
4x^{2} - 49 = 0
4^{2}x^{2} – 7^{2 }= 0
(4x)^{2 }– 7^{2 }= 0
(4x + 7)(4x - 7) = 0
4x + 7 = 0 or 4x - 7 = 0
x = -⁷⁄₄ or x = ⁷⁄₄
Example 10 :
(x - 2)^{2} - 36 = 0
Solution :
(x - 2)^{2} - 36 = 0
(x - 2)^{2} – 6^{2 }= 0
(x - 2 + 6)(x - 2 - 6) = 0
(x + 4)(x - 8) = 0
x + 4 = 0 or x - 8 = 0
x = -4 or x = 8
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