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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 through
(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 through $(2,9)$ . $\newline$ (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 through

(2,9)

.

\newline

(a) Find

k

.

\newline

(b) How many

x

-intercept(s) does the graph of

y=-g(x)+5

have?

Plug and Simplify Equation: Since the graph of $y=g(x)$ passes through $(2,9)$ , we plug $x=2$ and $y=9$ into the equation $g(x)=k^2x^2-10kx+6k+1$ . So, $9=k^2(2)^2-10k(2)+6k+1$ .
Combine Like Terms: Simplify the equation: $9=4k^2-20k+6k+1$ .
Set Equation to Zero: Combine like terms: $9=4k^2-14k+1$ .
Factor Quadratic Equation: Subtract $9$ from both sides to set the equation to zero: $0=4k^2-14k-8$ .
Solve for $k$ : Factor the quadratic equation: $(2k+1)(2k-8)=0$ .
Find X-Intercepts: Set each factor equal to zero and solve for $k$ : $2k+1=0$ or $2k-8=0$ .
Substitute $k$ into $g(x)$ : Solve the first equation: $2k+1=0$ leads to $k=-\frac{1}{2}$ , which we discard because $k$ is a positive constant.
Set Equation to Zero: Solve the second equation: $2k-8=0$ leads to $k=4$ .
Simplify Equation: Now we have the value of $k$ , which is $4$ . Next, we find the number of $x$ -intercepts for $y=-g(x)+5$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .Substitute $k=4$ into $g(x)$ to get $g(x)=4^2x^2-10(4)x+6(4)+1$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .Substitute $k=4$ into $g(x)$ to get $g(x)=4^2x^2-10(4)x+6(4)+1$ .Simplify $g(x)$ : $g(x)=16x^2-40x+25$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .Substitute $k=4$ into $g(x)$ to get $g(x)=4^2x^2-10(4)x+6(4)+1$ .Simplify $g(x)$ : $g(x)=16x^2-40x+25$ .Now set $-g(x)+5=0$ : $-16x^2+40x-25+5=0$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .Substitute $k=4$ into $g(x)$ to get $g(x)=4^2x^2-10(4)x+6(4)+1$ .Simplify $g(x)$ : $g(x)=16x^2-40x+25$ .Now set $-g(x)+5=0$ : $-16x^2+40x-25+5=0$ .Simplify the equation: $y=0$ $0$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .Substitute $k=4$ into $g(x)$ to get $g(x)=4^2x^2-10(4)x+6(4)+1$ .Simplify $g(x)$ : $g(x)=16x^2-40x+25$ .Now set $-g(x)+5=0$ : $-16x^2+40x-25+5=0$ .Simplify the equation: $y=0$ $0$ .Divide the equation by $y=0$ $1$ to make it simpler: $y=0$ $2$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .Substitute $k=4$ into $g(x)$ to get $g(x)=4^2x^2-10(4)x+6(4)+1$ .Simplify $g(x)$ : $g(x)=16x^2-40x+25$ .Now set $-g(x)+5=0$ : $-16x^2+40x-25+5=0$ .Simplify the equation: $y=0$ $0$ .Divide the equation by $y=0$ $1$ to make it simpler: $y=0$ $2$ .Use the discriminant $y=0$ $3$ to determine the number of $x$ -intercepts. Here, $y=0$ $5$ , $y=0$ $6$ , and $y=0$ $7$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .Substitute $k=4$ into $g(x)$ to get $g(x)=4^2x^2-10(4)x+6(4)+1$ .Simplify $g(x)$ : $g(x)=16x^2-40x+25$ .Now set $-g(x)+5=0$ : $-16x^2+40x-25+5=0$ .Simplify the equation: $y=0$ $0$ .Divide the equation by $y=0$ $1$ to make it simpler: $y=0$ $2$ .Use the discriminant $y=0$ $3$ to determine the number of $x$ -intercepts. Here, $y=0$ $5$ , $y=0$ $6$ , and $y=0$ $7$ .Calculate the discriminant: $y=0$ $8$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .Substitute $k=4$ into $g(x)$ to get $g(x)=4^2x^2-10(4)x+6(4)+1$ .Simplify $g(x)$ : $g(x)=16x^2-40x+25$ .Now set $-g(x)+5=0$ : $-16x^2+40x-25+5=0$ .Simplify the equation: $y=0$ $0$ .Divide the equation by $y=0$ $1$ to make it simpler: $y=0$ $2$ .Use the discriminant $y=0$ $3$ to determine the number of $x$ -intercepts. Here, $y=0$ $5$ , $y=0$ $6$ , and $y=0$ $7$ .Calculate the discriminant: $y=0$ $8$ .Simplify the discriminant: $y=0$ $9$ .
Calculate Discriminant: The $x$ -intercepts occur where $y=0$ . So we set $-g(x)+5=0$ .Substitute $k=4$ into $g(x)$ to get $g(x)=4^2x^2-10(4)x+6(4)+1$ .Simplify $g(x)$ : $g(x)=16x^2-40x+25$ .Now set $-g(x)+5=0$ : $-16x^2+40x-25+5=0$ .Simplify the equation: $y=0$ $0$ .Divide the equation by $y=0$ $1$ to make it simpler: $y=0$ $2$ .Use the discriminant $y=0$ $3$ to determine the number of $x$ -intercepts. Here, $y=0$ $5$ , $y=0$ $6$ , and $y=0$ $7$ .Calculate the discriminant: $y=0$ $8$ .Simplify the discriminant: $y=0$ $9$ .Since the discriminant is greater than $-g(x)+5=0$ $0$ , there are $-g(x)+5=0$ $1$ real and distinct $x$ -intercepts.

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