In the -plane, consider the system of equations where is a real constant. For how many distinct real values of does the system have exactly one solution ?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 209 | Free SAT Prep | Aniko

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Question 209·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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In the $xy$ -plane, consider the system of equations

y = (x - m)^2 + 1

y = mx

where $m$ is a real constant. For how many distinct real values of $m$ does the system have exactly one solution $(x,y)$ ?

For systems involving a line and a parabola with a parameter, set the equations equal to eliminate $y$ and obtain a quadratic in $x$ . Then use the discriminant $b^2 - 4ac$ to control the number of real intersections: set it $>0$ for two, $=0$ for exactly one, and $<0$ for none. Solve the resulting equation in the parameter, and remember that square roots give both positive and negative solutions when counting how many parameter values work.

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Relate the two equations

Because both expressions equal $y$ , try setting $(x - m)^2 + 1$ equal to $mx$ to get a single equation in $x$ and $m$ .

Notice the type of equation in x

After you set the equations equal and simplify, what kind of equation in $x$ do you get? Think about how many real solutions that type of equation can have.

Connect "exactly one solution" to the discriminant

For a quadratic in $x$ , how can you tell whether it has 0, 1, or 2 real solutions using the discriminant $b^2 - 4ac$ ?

Solve the discriminant condition for m

Write the discriminant of your quadratic in terms of $m$ , set it equal to zero, and solve for $m$ . Then decide how many distinct real $m$ values you found.

Desmos Guide

Graph both equations with a slider for m

Type y = (x - m)^2 + 1 and y = m x. When you enter m, Desmos will prompt you to add a slider; create the slider so you can change $m$ dynamically.

Use the slider to explore intersections

Move the $m$ -slider and watch how the line $y = mx$ moves relative to the parabola $y = (x - m)^2 + 1$ . Pay attention to how many intersection points appear for different $m$ values.

Identify when there is exactly one intersection point

Adjust $m$ until the line just touches the parabola at a single point (the graphs are tangent). Note each $m$ value where this happens. Count how many distinct $m$ values give exactly one intersection point.

Step-by-step Explanation

Set the two equations equal to each other

Since both equations equal $y$ , set the right-hand sides equal:

(x - m)^2 + 1 = mx

Now expand the square:

(x - m)^2 = x^2 - 2mx + m^2

So the equation becomes

x^2 - 2mx + m^2 + 1 = mx.

Rearrange into a standard quadratic in x

Move everything to one side to write the equation in standard quadratic form $ax^2 + bx + c = 0$ :

x^2 - 2mx + m^2 + 1 - mx = 0

Combine like terms:

x^2 - 3mx + m^2 + 1 = 0.

This is a quadratic in $x$ with

$a = 1$
$b = -3m$
$c = m^2 + 1$ .

The $x$ -solutions of this quadratic are exactly the $x$ -coordinates where the line and parabola intersect.

Use the discriminant for "exactly one" real solution

A quadratic $ax^2 + bx + c = 0$ has:

two distinct real solutions if $b^2 - 4ac > 0$
exactly one real solution if $b^2 - 4ac = 0$
no real solutions if $b^2 - 4ac < 0$ .

We want the system to have exactly one real solution $(x,y)$ , so we need the quadratic in $x$ to have exactly one real solution. That means its discriminant must be zero.

Compute the discriminant $D$ :

D = b^2 - 4ac = (-3m)^2 - 4(1)(m^2 + 1).

Simplify:

D = 9m^2 - 4m^2 - 4 = 5m^2 - 4.

Set this equal to zero:

5m^2 - 4 = 0.

Solve for m and count how many values work

Solve the equation from the discriminant condition:

5m^2 - 4 = 0 \\[0.7em] 5m^2 = 4 \\[0.7em] m^2 = \frac{4}{5}.

Taking square roots gives two real values:

m = \pm \frac{2\sqrt{5}}{5}.

These are two distinct real values of $m$ , and for each of them the quadratic in $x$ has exactly one real solution, so the system has exactly one solution $(x,y)$ . Therefore, the correct choice is "Two."

Strategy

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Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation