Question

questions 13 through 15 are gridded response items that require you to write your answers in the boxes provided on your answer sheet. write only one number or symbol in each box and fill in the circle in each column that matches what you have printed. fill in only one circle in each column.
13 what is the value of the positive zero of the function, f, defined by ( f(x) = x^2 - 121 )?
14 what is the value of x in the system of equations shown below?( 5x + 4y = 1 )( y = 1 - x )
15 what is the value of the smaller zero of the function ( f(x) = 2x^2 - 8x - 24 )?

Explanation:

Question 13

Step1: Set \( f(x) = 0 \)

To find the zeros of the function \( f(x)=x^{2}-121 \), we set \( f(x) = 0 \), so we have the equation \( x^{2}-121=0 \).

Step2: Solve for \( x \)

We can rewrite the equation as \( x^{2}=121 \). Taking the square root of both sides, we get \( x=\pm\sqrt{121} \). Since \( \sqrt{121} = 11 \), the solutions are \( x = 11 \) and \( x=- 11 \). The positive zero is \( 11 \).

Step1: Substitute \( y = 2 - x \) into \( 5x + 4y=1 \)

We have the system of equations:

$$
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Substitute \( y=2 - x \) into the first equation: \( 5x+4(2 - x)=1 \).

Step2: Simplify and solve for \( x \)

Expand the left - hand side: \( 5x + 8-4x=1 \).
Combine like terms: \( x+8 = 1 \).
Subtract 8 from both sides: \( x=1 - 8=-7 \).

Step1: Set \( f(x)=0 \)

To find the zeros of the function \( f(x)=2x^{2}-8x - 24 \), we set \( f(x) = 0 \), so \( 2x^{2}-8x - 24=0 \).

Step2: Simplify the equation

Divide the entire equation by 2: \( x^{2}-4x - 12=0 \).

Step3: Factor the quadratic equation

We need to find two numbers that multiply to \( - 12 \) and add up to \( - 4 \). The numbers are \( - 6 \) and \( 2 \). So, \( x^{2}-4x - 12=(x - 6)(x + 2)=0 \).

Step4: Solve for \( x \)

Set each factor equal to zero:

• If \( x - 6=0 \), then \( x = 6 \).
• If \( x+2=0 \), then \( x=-2 \).

The smaller zero is \( - 2 \).

Answer:

\( 11 \)

Question 14