First, we need to put the equation in standard form by:
1. Expanding the terms in parenthesis
2. Grouping like terms
3. Combining like terms
y = 2x^2 - 2x - 1 - (x - 7)^2
y = 2x^2 - 2x - 1 - (x^2 - 7x - 7x + 49)
y = 2x^2 - 2x - 1 - x^2 + 7x + 7x - 49
y = 2x^2 - x^2 - 2x+ 7x + 7x - 1 - 49
y = 2x^2 - 1x^2 - 2x+ 7x + 7x - 1 - 49
y = (2 - 1)x^2 + (-2+ 7 + 7)x + (-1 - 49)
y = 1x^2 + 12x + (-50)
y = 1x^2 + 12x - 50
We can now use the quadratic equation to solve this problem:
The quadratic formula states:
For color(red)(a)x^2 + color(blue)(b)x + color(green)(c) = 0, the values of x which are the solutions to the equation are given by:
x = (-color(blue)(b) +- sqrt(color(blue)(b)^2 - (4color(red)(a)color(green)(c))))/(2 * color(red)(a))
Substituting:
color(red)(1) for color(red)(a)
color(blue)(12) for color(blue)(b)
color(green)(-50) for color(green)(c) gives:
x = (-color(blue)(12) +- sqrt(color(blue)(12)^2 - (4 * color(red)(1) * color(green)(-50))))/(2 * color(red)(1))
x = (-12 +- sqrt(144 - (-200)))/2
x = (-12 +- sqrt(144 + 200))/2
x = (-12 +- sqrt(344))/2
x = (-12 +- sqrt(4 * 86))/2
x = (-12 +- sqrt(4)sqrt(86))/2
x = (-12 +- 2sqrt(86))/2
x = -6 +- sqrt(86)
