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Rewrite the function by completing the square.
f(x) = x^2 – 10x—96

Sagot :

Clarch19idn

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

Jacob193idn

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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