Super-Hard SAT Math Question 103 | Free SAT Prep | Aniko

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Question 103·200 Super-Hard SAT Math Questions·Advanced Math

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In the $xy$ -plane, a line and a parabola are given by the equations

\begin{aligned} \text{Line: } & y=x+1 \\ \text{Parabola: } & y=x^2-4x-5 \end{aligned}

If $(x,y)$ is an intersection point of the line and the parabola in the first quadrant, which choice is the value of $x$ ?

For a system containing a line and a parabola, set their two expressions for $y$ equal. This produces a quadratic equation whose solutions give the possible intersection $x$ -values. If the question specifies a quadrant or another location condition, substitute each candidate into an original equation and retain only the point that satisfies that condition.

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Use the intersection condition

At an intersection point, the two given expressions for $y$ must be equal.

Form a quadratic

Set the equations equal and move every term to one side. Then look for two numbers whose product is $-6$ and whose sum is $-5$ .

Check the location of each point

For each possible $x$ -value, use the line equation to find $y$ . A first-quadrant point must have both coordinates positive.

Desmos Guide

Graph both equations

Enter $y=x+1$ and $y=x^2-4x-5$ on separate expression lines in Desmos.

Identify the first-quadrant intersection

Click each intersection point. Select the point whose $x$ - and $y$ -coordinates are both positive.

Read the requested coordinate

Read the $x$ -coordinate of the first-quadrant point and match it to the answer choices.

Step-by-step Explanation

Set the expressions for $y$ equal

At an intersection point, both equations have the same $y$ -value. Therefore,

x+1=x^2-4x-5.

Solve for the possible $x$ -values

Move all terms to one side and factor:

\begin{aligned} 0&=x^2-5x-6 \\ 0&=(x-6)(x+1). \end{aligned}

Thus, the possible $x$ -values are $6$ and $-1$ .

Apply the first-quadrant condition

If $x=6$ , then $y=x+1=7$ , so $(6,7)$ is in the first quadrant. If $x=-1$ , then $y=0$ , which is not in the first quadrant.

Therefore, the correct choice is $6$ .

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