Generally we have two types of quadratic equation.
The general form of a quadratic equation is
ax2 + bx + c = 0
In a quadratic equation, leading coefficient is nothing but the coefficient of x2.
(i) In a quadratic equation in the form ax2 + 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 |
ax2 + bx + c = 0 |
Positive sign for both the factors. |
ax2 - bx + c = 0 |
Negative sign for both the factors. |
ax2 + bx - c = 0 |
Negative sign for smaller factor and positive sign for larger factor. |
ax2 - bx - c = 0 |
Positive sign for smaller factor and negative sign for larger factor. |
(i) In a quadratic equation in the form ax2 + bx + c = 0, if the leading coefficient is not 1, we have to multiply the coefficient of x2 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 x2 and simplify as much as possible.
(iv) Write the remaining number along with x (This is explained in the following example).
Example 1 :
Solve for x :
x2 + 9x + 14 = 0
Solution :
In the given quadratic equation, the coefficient of x2 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
+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 2 :
Solve for x :
x2 - 9x + 14 = 0
Solution :
In the given quadratic equation, the coefficient of x2 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
-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 3 :
Solve for x :
x2 + 2x - 15 = 0
Solution :
In the given quadratic equation, the coefficient of x2 is 1.
Decompose the constant term -15 into two factors such that the product of the two factors is equal to -15 and the addition of two factors is equal to the coefficient of x, that is +2.
Then, the two factors of -15 are
+5 and -3
Factor the given quadratic equation using +5 and -3 and solve for x.
(x + 5)(x - 3) = 0
x + 5 = 0 or x - 3 = 0
x = -5 or x = 3
So, the solution is {-5, 3}.
Example 4 :
Solve for x :
x2 - 2x - 15 = 0
Solution :
In the given quadratic equation, the coefficient of x2 is 1.
Decompose the constant term -15 into two factors such that the product of the two factors is equal to -15 and the addition of two factors is equal to the coefficient of x, that is -2.
Then, the two factors of -15 are
+3 and -5
Factor the given quadratic equation using +3 and -5 and solve for x.
(x + 3)(x - 5) = 0
x + 3 = 0 or x - 5 = 0
x = -3 or x = 5
So, the solution is {-3, 5}.
Example 5 :
Solve for x :
2x2 + 15x + 27 = 0
Solution :
In the given quadratic equation, the coefficient of x2 is not 1.
So, multiply the coefficient of x2 and the constant term "+27".
3 ⋅ (+27) = +54
Decompose +54 into two factors such that the product of two factors is equal to +54 and the addition of two factors is equal to the coefficient of x, that is +15.
Then, the two factors of +54 are
+6 and +9
Now we have to divide the two factors +6 and +9 by the coefficient of x2, that is 2.
(x + 3)(2x + 9) = 0
x + 3 = 0 or 2x + 9 = 0
x = -3 or x = -9/2
So, the solution is {-3, -9/2}.
Example 6 :
Solve for x :
2x2 - 15 x + 27 = 0
Solution :
In the given quadratic equation, the coefficient of x2 is not 1.
So, multiply the coefficient of x2 and the constant term "+27".
3 ⋅ (+27) = +54
Decompose +54 into two factors such that the product of two factors is equal to +54 and the addition of two factors is equal to the coefficient of x, that is +15.
Then, the two factors of +54 are
-6 and -9
Now we have to divide the two factors -6 and -9 by the coefficient of x2, that is 2.
(x - 3)(2x - 9) = 0
x - 3 = 0 or 2x - 9 = 0
x = 3 or x = 9/2
So, the solution is {3, 9/2}.
Example 7 :
Solve for x :
2x2 + 3x - 27 = 0
Solution :
In the given quadratic equation, the coefficient of x2 is not 1.
So, multiply the coefficient of x2 and the constant term "-27".
3 ⋅ (-27) = -54
Decompose -54 into two factors such that the product of two factors is equal to -54 and the addition of two factors is equal to the coefficient of x, that is +3.
Then, the two factors of -54 are
+9 and -6
Now we have to divide the two factors +9 and -6 by the coefficient of x2, that is 2.
(2x + 9)(x - 3) = 0
2x + 9 = 0 or x - 3 = 0
x = -9/2 or x = 3
So, the solution is {-9/2, 3}.
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