Consider the general form of the quadratic expression given below.
ax^{2} + bx + c
The following steps will be helpful to fctor a quadratic expression in the above form.
Step 1 :
Multiply the coefficient of x^{2}, a and the constant term c.
= ac
Step 2 :
Find two numbers p and q such that the product is equal to ac and the sum is equal to the coefficient of x, b.
pq = ac
p + q = b
Step 3 :
Split the middle term bx using the two numbers p and q.
= ax^{2} + px + qx + c
Step 4 :
Factor the expression in step 3 by grouping as shown below.
= (ax^{2} + px) + (qx + c)
Factor each of the following quadratic expressions.
Example 1 :
x^{2} + 5x + 6
Solution :
Step 1 :
In the quadratic expression x^{2} + 5x + 6, the coefficient of x^{2} is 1 and the constant term is 6.
Multiply 1 and 6.
= 1 x 6
= 6
Step 2 :
Find two numbers such that the product is equal to 6 and the sum is equal to the coeffient of x, 5.
The two numbers satisfy the above condition are 2 and 3.
Step 3 :
Split the middle term 5x using the two numbers 2 and 3.
= x^{2} + 2x + 3x + 6
Step 4 :
Factor the expression in the above step 3 by grouping.
= (x^{2} + 2x) + (3x + 6)
= x(x + 2) + 3(x + 2)
= (x + 2)(x + 3)
Example 2 :
x^{2} + 17x + 60
Solution :
Step 1 :
In the quadratic expression x^{2} + 17x + 60, the coefficient of x^{2 }is 1 and the constant term is 60.
Multiply 1 and 60.
= 1 x 60
= 60
Step 2 :
Find two numbers such that the product is equal to 60 and the sum is equal to the coeffient of x, 17.
The two numbers satisfy the above condition are 5 and 12.
Step 3 :
Split the middle term 5x using the two numbers 5 and 12.
= x^{2} + 5x + 12x + 60
Step 4 :
Factor the expression in the above step 3 by grouping.
= (x^{2} + 5x) + (12x + 60)
= x(x + 5) + 12(x + 5)
= (x + 5)(x + 12)
Example 3 :
x^{2} - 5x - 24
Solution :
Step 1 :
In the quadratic expression x^{2} - 5x - 24, the coefficient of x^{2 }is 1 and the constant term is 24.
Multiply 1 and -24.
= 1 x (-24)
= -24
Step 2 :
Find two numbers such that the product is equal to -24 and the sum is equal to the coeffient of x, -5.
The two numbers satisfy the above condition are -8 and 3.
Step 3 :
Split the middle term -5x using the two numbers -8 and 3.
= x^{2} - 8x + 3x - 24
Step 4 :
Factor the expression in the above step 3 by grouping.
= (x^{2} - 8x) + (3x - 24)
= x(x - 8) + 3(x - 8)
= (x - 8)(x + 3)
Example 4 :
x^{2} - 2x + 1
Solution :
Step 1 :
In the quadratic expression x^{2} - 2x + 1, the coefficient of x^{2 }is 1 and the constant term is 1.
Multiply 1 and 1.
= 1 x 1
= 1
Step 2 :
Find two numbers such that the product is equal to 1 and the sum is equal to the coeffient of x, -2.
The two numbers satisfy the above condition are -1 and -1.
Step 3 :
Split the middle term -2x using the two numbers -1 and -1.
= x^{2} - x - x + 1
Step 4 :
Factor the expression in the above step 3 by grouping.
= (x^{2} - x) + (-x + 1)
= x(x - 1) - 1(x - 1)
= (x - 1)(x - 1)
Example 5 :
2x^{2} + 11x + 12
Solution :
Step 1 :
In the quadratic expression 2x^{2} + 11x + 12, the coefficient of x^{2 }is 2 and the constant term is 12.
Multiply 2 and 12.
= 2 x 12
= 24
Step 2 :
Find two numbers such that the product is equal to 24 and the sum is equal to the coeffient of x, 11.
The two numbers satisfy the above condition are 3 and 8.
Step 3 :
Split the middle term 11x using the two numbers 3 and 8.
= 2x^{2} + 3x + 8x + 12
Step 4 :
Factor the expression in the above step 3 by grouping.
= (2x^{2} + 3x) + (8x + 12)
= x(2x + 3) + 4(2x + 3)
= (2x + 3)(x + 4)
Example 6 :
2x^{2} + x - 6
Solution :
Step 1 :
In the quadratic expression 2x^{2} + x - 6, the coefficient of x^{2} is 2 and the constant term is -6.
Multiply 2 and -6.
= 2 x (-6)
= -12
Step 2 :
Find two numbers such that the product is equal to -12 and the sum is equal to the coeffient of x, 1.
The two numbers satisfy the above condition are -3 and 4.
Step 3 :
Split the middle term x using the two numbers 3 and 8.
= 2x^{2} - 3x + 4x - 6
Step 4 :
Factor the expression in the above step 3 by grouping.
= (2x^{2} - 3x) + (4x - 6)
= x(2x - 3) + 2(2x - 3)
= (2x - 3)(x + 2)
Example 7 :
3x^{2} – 5x – 12
Solution :
Step 1 :
In the quadratic expression 3x^{2} – 5x – 12, the coefficient of x^{2 }is 3 and the constant term is -12.
Multiply 3 and -12.
= 3 x (-12)
= -36
Step 2 :
Find two numbers such that the product is equal to -36 and the sum is equal to the coeffient of x, -5.
The two numbers satisfy the above condition are -9 and 4.
Step 3 :
Split the middle term -5x using the two numbers -9 and 4.
= 3x^{2} – 9x + 4x – 12
Step 4 :
Factor the expression in the above step 3 by grouping.
= (3x^{2} – 9x) + (4x – 12)
= 3x(x - 3) + 4(x - 3)
= (x - 3)(3x + 4)
Example 8 :
x^{2} - 25
Solution :
The quadratic expression x^{2} – 25 can be factored using the following algebraic identity.
a^{2} - b^{2} = (a + b)(a - b)
= x^{2} – 25
= x^{2} – 5^{2}
= (x + 5)(x - 5)
Example 9 :
16x^{2} - 9
Solution :
= 16x^{2} - 9
= 4^{2}x^{2} – 3^{2}
= (4x)^{2 }– 3^{2}
= (4x + 3)(4x - 3)
Example 10 :
(x + 3)^{2} - 81
Solution :
= (x + 3)^{2} - 81
= (x + 3)^{2} – 9^{2}
= (x + 3 + 9)(x + 3 - 9)
= (x + 12)(x - 6)
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