Quadratic equations will be written in three forms.
(i) Standard form
(ii) Factored form
(iii) Vertex form
Here we see examples to show how to convert standard form to factored form.
The general form any quadratic equation will be in the form
ax^{2} + bx + c
To factorize a quadratic polynomial, we have to check whether the coefficient of x^{2} is 1 or not equal to 1.
If it is 1, take the constant alone and decompose it into two parts. Such that the product of those two terms should be equal to c and when we simplify, we should get b.
If it is not 1, take the constant alone and decompose it into two parts. Such that the product of those two terms should be equal to ac and when we simplify, we should get b.
Quadratic function y = ax^{2} + bx + c y = ax^{2} - bx + c y = ax^{2} - bx - c y = ax^{2} + bx - c |
Sign of factors Both factors are positive Both factors are negative The greater number will be positive, smaller number will be negative The greater number will be negative, smaller number will be positive |
Convert the following quadratic function from standard form to factored form.
Example 1 :
y = x^{2} - 4x + 4
Solution :
y = x^{2} - 4x + 4
Leading coefficient = 1 (coefficient of x^{2})
Constant = 4
Factors of 4 = -2 ⋅ (-2) and -2 + (-2) ==> -4
y = x^{2} - 2x - 2x + 4
Factoring x from first two terms and factoring 2 from third and fourth term.
y = x(x - 2) - 2(x - 2)
Factored form :
y = (x - 2) (x - 2)
Example 2 :
y = 2x^{2} - x - 1
Solution :
y = 2x^{2} - x - 1
Leading coefficient = 2 (coefficient of x^{2})
Constant = -1
Factors of -2 = -2 ⋅ 1 and -2 + 1 ==> -1
y = 2x^{2} - 2x + 1x - 1
y = 2x(x - 1) + 1(x - 1)
Factored form :
y = (2x + 1) (x - 1)
Example 3 :
y = x^{2} - 5x + 6
Solution :
y = x^{2} - 5x + 6
Leading coefficient = 1 (coefficient of x^{2})
Constant = -6
Factors of -6 = -2 ⋅ (-3) and -2 + (-3) ==> -5
y = x^{2} - 2x - 3x + 6
y = x(x - 2) - 3(x - 2)
Factored form :
y = (x - 2) (x - 3)
Example 4 :
y = x^{2} - 5x - 6
Solution :
y = x^{2} - 5x - 6
Leading coefficient = 1 (coefficient of x^{2})
Constant = -6
Factors of -6 = 1 ⋅ (-6) and 1 + (-6) ==> -5
y = x^{2} + 1x - 6x - 6
y = x(x + 1) - 6(x + 1)
Factored form :
y = (x - 6) (x + 1)
Example 5 :
y = x^{2} - 17x + 72
Solution :
y = x^{2} - 17x + 72
Leading coefficient = 1 (coefficient of x^{2})
Constant = 72
Factors of 72 = -8 ⋅ (-9) and -8 + (-9) ==> -17
y = x^{2} - 8x - 9x + 72
y = x(x - 8) - 9(x - 8)
Factored form :
y = (x - 8) (x - 9)
Example 6 :
y = 4x^{2} - 12x + 9
Solution :
y = 4x^{2} - 12x + 9
Leading coefficient = 4 (coefficient of x^{2})
Constant = 9
Factors of 36 = -6 ⋅ (-6) and -6 + (-6) ==> -12
y = 4x^{2} - 6x - 6x + 9
Factoring 2x from 4x^{2} - 6x and factoring -3 from - 6x + 9, we get
y = 2x(2x - 3) - 3(2x - 3)
Factored form :
y = (2x - 3) (2x - 3)
Example 7 :
y = 4x^{2} + 28x + 49
Solution :
y = 4x^{2} + 28x + 49
Leading coefficient = 4 (coefficient of x^{2})
Constant = 49
Factors of 196 = 14 ⋅ 14 and 14 + 14 ==> 28
y = 4x^{2} + 14x + 14x + 49
Factoring 2x from 4x^{2} + 14x and factoring 7 from 14x + 49, we get
y = 2x(2x + 7) + 7(2x + 7)
Factored form :
y = (2x + 7) (2x + 7)
Example 8 :
y = 9x^{2} + 6x + 1
Solution :
y = 9x^{2} + 6x + 1
Leading coefficient = 9 (coefficient of x^{2})
Constant = 1
Factors of 9 = 3 ⋅ 3 and 3 + 3 ==> 6
y = 9x^{2} + 3x + 3x + 1
Factoring 3x from 9x^{2} + 3x and factoring 1 from 3x + 1, we get
y = 3x(3x + 1) + 1(3x + 1)
Factored form :
y = (3x + 1) (3x + 1)
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