Solve a Quadratic Equation by Factoring :
In this section, you will learn how to solve quadratic equations by factoring.
Generally we have two types of quadratic equation.
The general form of a quadratic equation is
ax^{2} + bx + c = 0
In a quadratic equation, leading coefficient is nothing but the coefficient of x^{2}.
(i) In a quadratic equation in the form ax^{2} + bx + c = 0, 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 equation using p and q as shown below.
(x + p)(x + q) = 0
(iv) Solving the above equation, we get
x = -p and x = -q
Quadratic Equation |
Signs of Factors |
ax^{2} + bx + c = 0 |
Positive sign for both the factors. |
ax^{2} - bx + c = 0 |
Negative sign for both the factors. |
ax^{2} + bx - c = 0 |
Negative sign for smaller factor and positive sign for larger factor. |
ax^{2} - bx - c = 0 |
Positive sign for smaller factor and negative sign for larger factor. |
Example 1 :
Solve the following quadratic equation by factoring :
x^{2} + 9x + 14 = 0
Solution :
In the given quadratic equation, the coefficient of x^{2} is 1.
Decompose the constant term +14 into two factors such that the product of the two factors is equal to +14 and the addition of two factors is equal to the coefficient of x, that is +9.
Then, the two factors of +14 are
+7 and +2
Factor the given quadratic equation using +7 and +2 and solve for x.
(x + 7)(x + 2) = 0
x + 7 = 0 or x + 2 = 0
x = -7 or x = -2
So, the solution is {-7, -2}.
Example 2 :
Solve the following quadratic equation by factoring :
x^{2} - 9x + 14 = 0
Solution :
In the given quadratic equation, the coefficient of x^{2} is 1.
Decompose the constant term +14 into two factors such that the product of the two factors is equal to +14 and the addition of two factors is equal to the coefficient of x, that is -9.
Then, the two factors of +14 are
-7 and -2
Factor the given quadratic equation using -7 and -2 and solve for x.
(x - 7)(x - 2) = 0
x - 7 = 0 or x - 2 = 0
x = 7 or x = 2
So, the solution is {7, 2}.
Example 3 :
Solve the following quadratic equation by factoring :
x^{2} + 5x - 14 = 0
Solution :
In the given quadratic equation, the coefficient of x^{2} is 1.
Decompose the constant term -14 into two factors such that the product of the two factors is equal to -14 and the addition of two factors is equal to the coefficient of x, that is +5.
Then, the two factors of -14 are
+7 and -2
Factor the given quadratic equation using +7 and -2 and solve for x.
(x + 7)(x - 2) = 0
x + 7 = 0 or x - 2 = 0
x = -7 or x = 2
So, the solution is {-7, 2}.
Example 4 :
Solve the following quadratic equation by factoring :
x^{2} - 5x - 14 = 0
Solution :
In the given quadratic equation, the coefficient of x^{2} is 1.
Decompose the constant term -14 into two factors such that the product of the two factors is equal to -14 and the addition of two factors is equal to the coefficient of x, that is -5.
Then, the two factors of -14 are
+2 and -7
Factor the given quadratic equation using +2 and -7 and solve for x.
(x + 2)(x - 7) = 0
x + 2 = 0 or x - 7 = 0
x = -2 or x = 7
So, the solution is {-2, 7}.
Example 5 :
Solve the quadratic equation by factoring :
3x^{2} – 5x – 12 = 0
Solution :
In the given quadratic equation, 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.
Now, factor the given quadratic equation and solve for x as shown below.
(3x + 4)(x - 3) = 0
3x + 4 = 0 or x - 3 = 0
x = -4/3 or x = 3
So, the solution is {-4/3, 3}.
After having gone through the stuff given above, we hope that the students would have understood how to solve quadratic equations by factoring
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