First, expand the term squared on the right side of the equation using the rule:
#(color(red)(a) + color(blue)(b))^2 = color(red)(a)^2 + 2color(red)(a)color(blue)(b) + color(blue)(b)^2#
Substituting #x# for #a# and #2# for #b# gives:
#y = -2(color(red)(x) + color(blue)(2))^2 + 16x^2 - x - 5#
#y = -2(color(red)(x)^2 + (2 * color(red)(x) * color(blue)(2)) + color(blue)(2)^2) + 16x^2 - x - 5#
#y = -2(x^2 + 4x + 4) + 16x^2 - x - 5#
Now, expand the term in parenthesis, group and combine like items:
#y = (-2 * x^2) + (2 * 4x) + (2 * 4) + 16x^2 - x - 5#
#y = -2x^2 + 8x + 8 + 16x^2 - x - 5#
#y = -2x^2 + 16x^2 + 8x - x + 8 - 5#
#y = (-2 + 16)x^2 + (8 - 1)x + (8 - 5)#
#y = 14x^2 + 7x + 3#
We can now use the quadratic equation to solve this problem. The quadratic formula states:
For #ax^2 + bx + 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)(14)# for #color(red)(a)#
#color(blue)(7)# for #color(blue)(b)#
#color(green)(3)# for #color(green)(c)# gives:
#x = (-color(blue)(7) +- sqrt(color(blue)(7)^2 - (4 * color(red)(14) * color(green)(3))))/(2 * color(red)(14))#
#x = (-color(blue)(7) +- sqrt(49 - 168))/28#
#x = (-color(blue)(7) +- sqrt(-119))/28#
Or
#x = -color(blue)(7)/28 +- sqrt(-119)/28#
#x = -1/4 +- sqrt(-119)/28#
