Super-Hard SAT Math Question 110 | Free SAT Prep | Aniko

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Question 110·200 Super-Hard SAT Math Questions·Advanced Math

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x^3 + (b-2)x^2 - (2b+15)x - 15b = 0

In the given equation, $b$ is a positive integer constant. If the equation has exactly two distinct real solutions, which choice could be a solution to the equation? anikо.‍аi

When a polynomial contains a parameter like $b$ , first try factoring by grouping: separate the $b$ -terms from the non- $b$ terms and look for a shared factor. After factoring, list the roots in terms of $b$ . If the problem adds a condition like “exactly two distinct real solutions,” interpret it as “one root must repeat,” meaning two factors must produce the same root. Finally, use any constraint on $b$ (such as $b>0$ and integer) to narrow down which roots can coincide and which answer choice fits. ani‍kо.aі

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Group terms with and without $b$

Try rewriting the polynomial as something like (terms without $b$ ) plus $b$ (terms). Look for a common factor you can pull out.

Think about what “exactly two distinct solutions” implies

If a factored equation would normally give three roots, having only two distinct real solutions means one of the roots must repeat (two factors give the same root). (Аnікο.а⁠і)

Use the fact that $b$ is positive

Once you identify a root that looks like $-b$ , remember it must be negative because $b>0$ . That rules out matching it with any positive root.

Desmos Guide

Create a slider for $b$

Enter a function with a parameter:

f(x)=x^3+(b-2)x^2-(2b+15)x-15b

Desmos will create a slider for $b$ . аniко.ai

Restrict $b$ to positive integers

In the slider settings, set $b$ to start at 1 and increase by 1 (step size 1).

Look for the case with exactly two distinct x-intercepts

Move the slider for $b$ and watch the graph of $y=f(x)$ . Find a value of $b$ where the graph has only two distinct x-intercepts (one of them will look like the graph just touches the x-axis and turns around, indicating a repeated root).

Match the repeated/root x-value to the choices

Click the x-intercepts to see their x-coordinates, then compare the x-values you observe to the answer choices. Choose the option that matches an intercept when the graph has exactly two distinct x-intercepts.

Step-by-step Explanation

Factor by grouping

Rewrite the polynomial by grouping the terms with and without $b$ : S‌AT prер bу⁠ Anіko.а⁠і

\begin{aligned} 0&=x^3 + (b-2)x^2 - (2b+15)x - 15b \\ &=(x^3-2x^2-15x) + b(x^2-2x-15) \\ &=x(x^2-2x-15) + b(x^2-2x-15) \end{aligned}

Factor out the common expression

Both groups contain $(x^2-2x-15)$ :

(x^2-2x-15)(x+b)=0

Also factor the quadratic:

x^2-2x-15=(x-5)(x+3)

List the possible solutions

From

(x-5)(x+3)(x+b)=0,

the real solutions are:

$x=5$
$x=-3$
$x=-b$

Use “exactly two distinct” to determine what must happen

Normally there are 3 distinct real solutions, but the problem says there are exactly 2 distinct real solutions. That means two of the three roots must be the same.

Since $b$ is a positive integer, $x=-b$ is negative. It cannot equal $5$ , so it must equal $-3$ :

-b=-3 \Rightarrow b=3.

Then the solutions are $x=-3$ (repeated) and $x=5$ , so a value that could be a solution is $-3$ .

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