Question 11·25 Super-Hard SAT Advanced Math Questions·Advanced Math

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In the $xy$ -plane, the line $y=mx+b$ , where $m>0$ , intersects the parabola $y=x^2-4x+1$ at two distinct points. The difference between the $x$ -coordinates of the two points is $8$ , and the sum of their $y$ -coordinates is $44$ . What is the value of $m$ ?

Your answer

A line meeting a parabola twice gives two roots when you set their $y$ -expressions equal. Express the intersection $x$ -values as a midpoint plus or minus half their gap; their sum connects the midpoint to the line’s slope. A restricted Desmos regression handles the remaining $y$ -sum. Without the slope’s sign condition, it could find the wrong line.

Hints

Hint 1
An intersection lies on both graphs, so set their $y$ -expressions equal. The resulting quadratic’s roots, or solutions, are the intersection $x$ -coordinates. Which coefficient contains the unknown slope?
Hint 2
Use a midpoint $h$ for the two $x$ -coordinates. A gap of $8$ puts them $4$ to either side of $h$ . What is their sum?
Hint 3
The parabola gives both $y$ -coordinates. Add its outputs at the two midpoint-based $x$ -values, then solve the sum condition in Desmos. Keep the restriction that the line’s slope is positive.

Step-by-step

Use the roots’ sum, then solve the $y$ -sum
Step 1
Make an equation for the intersection coordinates
At an intersection, the line and parabola give the same $y$ . Set their expressions equal: $x^2-4x+1=mx+b$ . Move every term to the left: $x^2-(m+4)x+(1-b)=0$ . The two intersection $x$ -coordinates are this quadratic’s roots, the values that make it zero.
Step 2
Write the coordinates around their midpoint
Call the midpoint of the two $x$ -coordinates $h$ . They are $8$ apart, so each is $4$ from the midpoint. Write them as: $x_1=h-4,\qquad x_2=h+4$ . Add them: $x_1+x_2=2h$ . The $8$ is the gap, not either coordinate.
Step 3
Connect the midpoint to the slope
For $x^2+px+q=0$ , the roots add to $-p$ : multiplying $(x-x_1)(x-x_2)$ makes its middle term $-(x_1+x_2)x$ . Here the middle coefficient is $-(m+4)$ , so the roots’ sum is: $x_1+x_2=m+4$ . Use the sum from the midpoint: $2h=m+4$ . Subtract $4$ to isolate the requested slope:
$m=2h-4.$
Step 4
Keep the positive-slope condition
The problem says $m>0$ . Put that condition into $m=2h-4$ : $2h-4>0$ . Add $4$ : $2h>4$ . Divide by $2$ : $h>2$ . This restriction will rule out a midpoint that produces a negative slope.
Step 5
Find the midpoint from the two $y$ -coordinates
Define the parabola’s output as $f(x)=x^2-4x+1$ in Desmos. Then type $f(h_1-4)+f(h_1+4)\sim44\{h_1>2\}$ . Here $h_1$ stands for the midpoint, and $\sim$ asks Desmos to find it from the sum of the two outputs. Under PARAMETERS, Desmos shows $h_1=5$ .
Step 6
Convert the midpoint to the slope
The midpoint is not the slope. Add $2h_1-4$ in Desmos; it prints $6$ . So the line’s slope is $m=6$ . Grid in 6.

Question 11·25 Super-Hard SAT Advanced Math Questions·Advanced Math

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