Question

to practice her programming skills, kaitlyn wrote a program that takes an input number, squares it, adds a constant value, multiplies the result by another constant value, and then outputs the result.
when the input is 8, the output is 204. when the input is 3, the output is 39.
what is the output when the input is 5?
show your work and solution below.
explain your process in complete sentences.

Explanation:

Step1: Define the formula

Let the input number be \( x \), the first constant be \( a \), and the second constant be \( b \). The formula for the output \( y \) is \( y = b(x^2 + a) \).

Step2: Substitute input 8 and output 204

When \( x = 8 \) and \( y = 204 \), we have \( 204 = b(8^2 + a) \), which simplifies to \( 204 = b(64 + a) \).

Step3: Substitute input 3 and output 39

When \( x = 3 \) and \( y = 39 \), we have \( 39 = b(3^2 + a) \), which simplifies to \( 39 = b(9 + a) \).

Step4: Solve the system of equations

We now have a system of equations:
\[

$$\begin{cases} 204 = b(64 + a) \\ 39 = b(9 + a) \end{cases}$$

\]
From the second equation, we can express \( b \) as \( b = \frac{39}{9 + a} \). Substitute this into the first equation:
\[
204 = \frac{39}{9 + a}(64 + a)
\]
Cross - multiply: \( 204(9 + a)=39(64 + a) \)
Expand both sides: \( 1836+204a = 2496+39a \)
Subtract \( 39a \) from both sides: \( 1836 + 165a=2496 \)
Subtract 1836 from both sides: \( 165a=2496 - 1836=660 \)
Divide both sides by 165: \( a=\frac{660}{165} = 4 \)

Step5: Find the value of \( b \)

Substitute \( a = 4 \) into \( b=\frac{39}{9 + a} \), we get \( b=\frac{39}{9 + 4}=\frac{39}{13}=3 \)

Step6: Find the output when \( x = 5 \)

Now that we know \( a = 4 \) and \( b = 3 \), the formula is \( y = 3(x^2+4) \). When \( x = 5 \), we have \( y=3(5^2 + 4)=3(25 + 4)=3\times29 = 87 \)

Answer:

The output when the input is 5 is 87.