Factoring Quadratic Expressions :
In this section, you will learn how to factor quadratic expressions.
Generally we have two types of quadratic expression.
They are
The general form of a quadratic expression is
ax^{2} + bx + c
In a quadratic expression, leading coefficient is nothing but the coefficient of x^{2}.
(i) In a quadratic expression in the form ax^{2} + bx + c, if the leading coefficient is 1, we have to decompose the constant term "c" into two factors.
(ii) The product of the two factors must be equal to the constant term "c" and the addition of two factors must be equal to the coefficient of x, that is "b".
(iii) If p and q are the two factors of the constant term c, then we have to factor the quadratic expression using p and q as shown below.
(x + p)(x + q)
Quadratic Expression |
Signs of Factors |
ax^{2} + bx + c |
Positive sign for both the factors. |
ax^{2} - bx + c |
Negative sign for both the factors. |
ax^{2} + bx - c |
Negative sign for smaller factor and positive sign for larger factor. |
ax^{2} - bx - c |
Positive sign for smaller factor and negative sign for larger factor. |
Example :
Factor :
x^{2} + 17 x + 60
Solution :
The given quadratic expression is in the form of
ax^{2} + bx + c
Check whether the coefficient of x^{2} is 1 or not.
Because the coefficient of x^{2} is 1, we have to decompose 60 into two factors as shown below.
Because the constant term 60 is having positive sign, both the factors must be positive.
In the above four pairs of factors, we have to select the a pair of factors such that the product of two factors is equal to the constant term "+60" and the addition of two factors is equal to the coefficient of x, that is "+17".
Now, factor the given quadratic expression as shown below.
x^{2} + 17 x + 60 = (x + 12)(x + 5)
Therefore, the factors of the given quadratic expression are
(x + 12) and (x + 5)
(i) In a quadratic expression in the form ax^{2} + bx + c, if the leading coefficient is not 1, we have to multiply the coefficient of x^{2} and the constant term. That is "ac". Then, decompose "ac" into two factors.
(ii) The product of the two factors must be equal to "ac" and the addition of two factors must be equal to the coefficient of x, that is "b".
(iii) Divide the two factors by the coefficient of x^{2} and simplify as much as possible.
(iv) Write the remaining number along with x (This is explained in the following example).
Factor :
2x^{2} + x - 6
Solution :
The given quadratic expression is in the form of
ax^{2} + bx + c
Here, the coefficient of x^{2} is 1 or not.
Multiply the coefficient of x^{2} and the constant term "-6".
That is,
2 ⋅ (-6) = -12
Decompose -12 into two factors such that the product of two factors is equal to -12 and the addition of two factors is equal to the coefficient of x, that is 1.
Then, the two factors of -12 are
4 and -3
Now we have to divide the two factors 4 and -3 by the coefficient of x^{2}, that is 2.
2x^{2} + x - 6 = (x + 2)(2x - 3)
Therefore, the factors of the given quadratic expression are
(x + 2) and (2x - 3)
Problem 1 :
Factor :
x^{2} – 5x – 24
Solution :
In the given quadratic expression, the coefficient of x^{2} 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 expression using +3 and -8.
x^{2} – 5x – 24 = (x + 3)(x - 8)
Therefore, the factors of the given quadratic expression are
(x + 3) and (x - 8)
Problem 2 :
Factor :
3x^{2} – 5x – 12
Solution :
In the given quadratic expression, the coefficient of x^{2} is not 1.
So, multiply the coefficient of x^{2} 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 -3 by the coefficient of x^{2}, that is 3.
3x^{2} – 5x – 12 = (3x + 4)(x - 3)
Therefore, the factors of the given quadratic expression are
(3x + 4) and (x - 3)
Problem 3 :
Factor :
(x + 3)^{2} - 81
Solution :
(x + 3)^{2} - 81 = (x + 3)^{2} - 9^{2}
In the above equality, use the algebraic identity
a^{2} - b^{2} = (a + b)(a - b)
on the right side and factor.
Then,
(x + 3)^{2} - 81 = [(x + 3) + 9][(x + 3) - 9]
(x + 3)^{2} - 81 = [x + 3 + 9][x + 3 - 9]
(x + 3)^{2} - 81 = (x + 12)(x - 6)
Therefore, the factors of the given quadratic expression are
(x + 12) and (x - 6)
After having gone through the stuff given above, we hope that the students would have understood how to factor quadratic expressions.
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