Super-Hard SAT Advanced Math Question 14 | Free SAT Prep

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Question 14·25 Super-Hard SAT Advanced Math Questions·Advanced Math

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A quadratic function $q$ is defined by

q(x)=ax^2+bx+c,

where $a$ is a positive integer. The function $q$ has two distinct integer zeros and satisfies the following conditions:

\begin{aligned} q(3)&>0 \\ q(5)&<0 \\ q(10)&>0 \\ q(0)&=72 \\ q(6)&=-12 \end{aligned}

What is the value of $b$ ?

Your answer

For a quadratic with integer zeros, use positive and negative function values to place the zeros before doing any coefficient algebra. Once the zeros are represented in factored form, exact function values create equations involving the leading coefficient and any unknown zero. Solve those equations, then use the expanded factored form to identify the requested coefficient.

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Use the signs

Because the parabola opens upward, determine where each zero must lie by comparing the signs at $x=3$ , $x=5$ , and $x=10$ .

Name the second zero

After identifying the first zero, call the other zero $s$ and write $q(x)$ in factored form using $a$ , $4$ , and $s$ .

Use both exact function values

Substitute $x=0$ and $x=6$ into the factored form. The resulting equations can be combined to determine both $a$ and $s$ .

Desmos Guide

Use algebra first

Algebra is faster for locating the integer zero at $4$ from the sign changes. Then write the remaining zero as $s$ .

Graph the two relationships

From the function values, the relationships are $as=18$ and $a(s-6)=6$ . In Desmos, treat the horizontal coordinate as $a$ and the vertical coordinate as $s$ . Graph

y=\frac{18}{x}

and

y=6+\frac{6}{x}.

Their positive intersection gives the values of $a$ and $s$ .

Calculate the coefficient

Use $b=-a(4+s)$ with the intersection values to verify the requested coefficient.

Step-by-step Explanation

Locate the zeros

Because $a$ is positive, the graph opens upward. Since $q(3)>0$ and $q(5)<0$ , one zero must be between $3$ and $5$ . Because the zeros are integers, that zero is $4$ .

Since $q(5)<0$ and $q(10)>0$ , the other zero, call it $s$ , is an integer between $5$ and $10$ .

Write the factored form

The function can be written as

q(x)=a(x-4)(x-s).

Using $q(0)=72$ gives

4as=72,

as=18.

Using $q(6)=-12$ gives

2a(6-s)=-12,

a(s-6)=6.

Find the leading coefficient and second zero

Rewrite the second equation as $as-6a=6$ . Since $as=18$ ,

18-6a=6.

Thus $a=2$ , and then $as=18$ gives $s=9$ .

Find the coefficient of $x$

Expand only the part needed for $b$ :

q(x)=a\left(x^2-(4+s)x+4s\right).

Therefore,

b=-a(4+s)=-2(4+9)=-26.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation