The solutions are the roots of the equation and can be found by finding the x-intercepts or zeros of the quadratic function. In this case the roots are: x= 1, and x = 2.
Certain equations have double roots. For example, the equation x2 + 6x + 9 = 0 has roots x = - 3, and x = - 3. Here is the plot:
The x - intercept is x = - 3.
By Finding the Square Root:
x2 + 6x + 9 = 8
(x + 3)2 = 8
Take the square root:
(x + 3) = √8
x = - 3 ± √8 = - 3 ± 2.83
x ≈ - 5.83, x = - 0.17
By Completing the Square:
Can only be used if the coefficient of x2 is 1.
x2 - 6x + 4 = 11
x2 - 6x = 11 - 4 = 7
Divide the coefficient of x by 2: 6/2 = 3
Square the result: 3 x 3 = 9
Add that result to both sides:
x2 - 6x + 9 = 7 + 9 = 16
(x - 3)2 = 16
Take the square root of both sides:
(x - 3) = ± 4
x = ± 4 + 3
x = 7, x = -1
If the coefficient of x2 does not equal 1, then divide both sides of the equation by the coefficient of x2 .
By Using the Quadratic Formula:
The solution of the standard form of a quadratic function is: ax2 + bx + c, where a ≠ 0 can be found using the Quadratic Formula:
x = - b ± √b2 - 4 ac
2a
Solve: x2 - 4x -32 = 0
a = 1; b = - 4; c = - 32
x = - ( -4) ± √(- 4)2 - 4 (1)(- 32) = 4 ± √ 16- (-128) = 4 ± √ 144 = 4 ±12 = 2 ± 6
2(1) 2 2 2
x = 8; x = - 4