Answer:
x = -1 ± (√6)/2
Step-by-step explanation:
You want the solution to 8x^2+16x-4=0 two ways.
Reducing the equation
We notice all of the coefficients are multiples of 4. This means we can divide that factor out, leaving us with the quadratic ...
2x^2 +4x -1 = 0
Completing the square
One way to solve such a quadratic is by "completing the square." The solutions are found by taking the square root of the square.
2x^2 +4x -1 = 0
2x^2 +4x = 1 . . . . . . . separate the constant from the variable terms
2(x^2 +2x) = 1 . . . . . . factor out the leading coefficient
2(x^2 +2x +1) = 1 +2(1) . . . . . add the square of half the x coefficient
The square is completed at this point.
2(x +1)^2 = 3 . . . . . . . . write the left side as a square, collect terms
(x +1)^2 = 3/2 . . . . . . . . divide by the leading coefficient
x +1 = ±√(3/2) = ±(√6)/2 . . . . . take the square root
x = -1 ± (√6)/2 . . . . . . . subtract 1; the solutions to the equation
Quadratic formula
The reduced equation can be compared to the standard form ...
ax^2 +bx +c = 0
to see that a=2, b=4, c=-1.
The solution to the equation using the quadratic formula is ...
[tex]x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\\\\x=\dfrac{-4\pm\sqrt{4^2-4(2)(-1)}}{2(2)}\\\\\boxed{x=-1\pm\dfrac{\sqrt{6}}{2}}[/tex]
Graph
The attached graph shows the solutions to be approximately -2.225 and +0.225. The vertex (-1, -12) and the leading coefficient (8) tell you these solutions are -1±√(12/8) = -1±(√6)/2.
