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Let
g(x)=k^(2)x^(2)-10 kx+6k+1, where
k is a positive constant. The graph of
y=g(x) passes fhrough
(2,9).
(a) Find
k.
(b) How many
x-intercept(s) does the graph of
y=-g(x)+5 have?

$7$ . Let $g(x)=k^{2} x^{2}-10 k x+6 k+1$ , where $k$ is a positive constant. The graph of $y=g(x)$ passes fhrough $(2,9)$ . $\newline$ (a) Find $k$ . $\newline$ (b) How many $x$ -intercept(s) does the graph of $y=-g(x)+5$ have?

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Conjugate root theorems

Full solution

Q.

7

. Let

g(x)=k^{2} x^{2}-10 k x+6 k+1

, where

k

is a positive constant. The graph of

y=g(x)

passes fhrough

(2,9)

.

\newline

(a) Find

k

.

\newline

(b) How many

x

-intercept(s) does the graph of

y=-g(x)+5

have?

Use Quadratic Formula: Use the quadratic formula to solve for $k$ . $k = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(4)(-8)}}{2(4)}$ $k = \frac{14 \pm \sqrt{196 + 128}}{8}$ $k = \frac{14 \pm \sqrt{324}}{8}$ $k = \frac{14 \pm 18}{8}$
Calculate Possible Values: Calculate the two possible values for $k$ . $k = \frac{14 + 18}{8}$ or $k = \frac{14 - 18}{8}$ $k = \frac{32}{8}$ or $k = \frac{-4}{8}$ $k = 4$ or $k = -0.5$ Since $k$ is a positive constant, $k = 4$ .
Substitute $k = 4$ : Substitute $k = 4$ into $g(x)$ to find the x-intercepts of $y = -g(x) + 5$ .
$g(x) = 4^2x^2 - 10(4)x + 6(4) + 1$
$g(x) = 16x^2 - 40x + 24 + 1$
$g(x) = 16x^2 - 40x + 25$
$y = -g(x) + 5 = -16x^2 + 40x - 25 + 5$
$y = -16x^2 + 40x - 20$
Determine X-Intercepts: Determine the number of x-intercepts by finding the discriminant of $y = -16x^2 + 40x - 20$ . $\newline$ Discriminant, $D = b^2 - 4ac$ $\newline$ $D = (40)^2 - 4(-16)(-20)$ $\newline$ $D = 1600 - 1280$ $\newline$ $D = 320$
Discriminant Analysis: Since the discriminant $D > 0$ , there are two distinct real $x$ -intercepts.

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