The equation of a parabola given [tex] y=-x^2-14x-59 [/tex]
We have to write it in vertex form.
The vertex form of a parabola is [tex] y=a(x-h)^2+k [/tex]
So to write it in vertex form we have to make the right side as a perfect square by using completing the square method.
[tex] y= -x^2-14x-59 [/tex]
[tex] y=-(x^2+14x+59) [/tex]
Here 14x given. By dividing 14 by 2 we will get 7. So we have to add and subtract [tex] (7)^2 [/tex] to the right side.
[tex] y= -[x^2+14x+(7)^2-(7)^2+59] [/tex]
[tex] y=-[x^2+14x+(7)^2-49+59] [/tex]
We know that, [tex] a^2+2ab+b^2 = (a+b)^2 [/tex], so [tex] x^2+(2)(x)(7)+7^2 = (x+7)^2 [/tex].
[tex] y= -[(x+7)^2-49+59] [/tex]
[tex] y=-[(x+7)^2 +10] [/tex]
[tex] y=-(x+7)^2-10 [/tex]
So we have got the required vertex form of the parabola.
