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Sagot :
Answer:
(x - 5)² -121 = 0
Step-by-step explanation:
if you need to find the roots you can take the square root of each side:
(x-5)² = 121
(x-5)² = 121
square root of (x-5)² is x-5
square root of 121 is ±11
first root: x-5 = 11
x = 16
second root: x-5 = -11
x = -6
Answer:
[tex]f(x) = (x - 5)^{2} - 121[/tex].
Step-by-step explanation:
The goal is to rewrite [tex]f(x)[/tex] in the vertex form [tex]a\, (x - h)^{2} + k[/tex] by completing the square (where [tex]a[/tex], [tex]h[/tex], and [tex]k[/tex] are constants.)
Expand the vertex form expression:
[tex]\begin{aligned}& a\, (x - h)^{2} + k\\ &= a\, (x - h)\, (x - h) + k \\ &= a\, \left(x^2 - h\, x - h\, x + h^2\right) + k \\ &= a\, \left(x^2 - 2\, h\, x + h^2\right) + k\\ &= a\, x^2 - 2\, a\, h\, x + \left(a\, h^2 + k\right) \end{aligned}[/tex].
Compare this expression to [tex]f(x) = x^2 - 10\, x - 96[/tex] and solve for the constants [tex]a[/tex], [tex]h[/tex], and [tex]k[/tex]. Make sure that the coefficient of each term matches:
- Coefficient for the [tex]x^2[/tex] term: [tex]a[/tex] in the expanded expression and [tex]1[/tex] in the expression for [tex]f(x)[/tex]. Hence, [tex]a = 1[/tex].
- Coefficient for the [tex]x[/tex] term: [tex](-2\, a\, h)[/tex] in the expanded expression and [tex](-10)[/tex] in the expression for [tex]f(x)[/tex]. Hence, [tex]-2\, a\, h = -10[/tex].
- Coefficient for the constant term: [tex]\left(a\, h^2 + k\right)[/tex] in the expanded expression and [tex](-96)[/tex] in the expression for [tex]f(x)[/tex]. Hence, [tex]a\, h^{2} + k = -96[/tex].
Substitute [tex]a = 1[/tex] into the second equation, [tex]-2\, a\, h = -10[/tex], and solve for [tex]h[/tex].
[tex]-2 \, h = -10[/tex].
[tex]h = 5[/tex].
Substitute both [tex]a = 1[/tex] and [tex]h = 5[/tex] into the third equation, [tex]a\, h^{2} + k = -96[/tex], and solve for [tex]k[/tex].
[tex]5^2 + k = -96[/tex].
[tex]k = -121[/tex].
Therefore, [tex]a\, (x - h)^{2} + k[/tex] becomes [tex](x - 5)^2 + (-121)[/tex].
Hence, the vertex form of the parabola [tex]f(x)[/tex] would be:
[tex]f(x) = (x - 5)^{2} - 121[/tex].
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