Let be a real number. Consider the system of equations For which value of does the system have exactly one real solution?

Solution available with step-by-step explanation

SAT Nonlinear Equations & Systems Question 53 | Free SAT Prep | Aniko

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Question 53·Medium·Nonlinear Equations in One Variable; Systems in Two Variables

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Let $b$ be a real number. Consider the system of equations

y = -x^2 + 4x + 3

y = 2x + b

For which value of $b$ does the system have exactly one real solution?

When a system involves a line and a parabola and asks for the number of real solutions, think about intersections: usually there are 0, 1, or 2. Set the two expressions for $y$ equal to get a quadratic in $x$ , then use the discriminant $D = b^2 - 4ac$ to control the number of real solutions: $D > 0$ gives two, $D = 0$ gives exactly one, and $D < 0$ gives none. On the SAT, go straight to writing the quadratic and discriminant in terms of the parameter, set $D$ to the required value (often 0 for “exactly one solution”), and solve for the parameter instead of trying to reason purely from the graph.

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Connect one solution to the graph shapes

You are intersecting a parabola and a line. How many intersection points do they usually have, and what must be special about the line for there to be exactly one intersection point?

Turn the system into a single equation

Set the two expressions for $y$ equal to each other to get one equation in terms of $x$ and $b$ . Then rearrange it into the standard quadratic form $ax^2 + bx + c = 0$ .

Use the discriminant condition

For the quadratic you found, write the discriminant $D = b^2 - 4ac$ (using the coefficients of $x^2$ , $x$ , and the constant term). What value must $D$ have for there to be exactly one real solution? Set up and solve that equation in terms of $b$ .

Desmos Guide

Graph the parabola

In Desmos, enter the parabola equation: y = -x^2 + 4x + 3.

Test each answer choice as a line

For each option $b = 0, 2, 4, 6$ , type the corresponding line into Desmos (for example, y = 2x + 0, y = 2x + 2, etc.). After typing each line, look at how many intersection points it has with the parabola (Desmos will show intersection points when you tap on the curves).

Identify the special case

Compare the four lines: find the value of $b$ for which the line just touches the parabola at exactly one point (the line is tangent). That value of $b$ is the one that makes the system have exactly one real solution.

Step-by-step Explanation

Set the two equations equal

For a solution to the system, both expressions for $y$ must be equal:

-x^2 + 4x + 3 = 2x + b.

Move all terms to one side:

-x^2 + 4x - 2x + 3 - b = 0 \\[0.7em] -x^2 + 2x + 3 - b = 0.

Multiply both sides by $-1$ to make the $x^2$ coefficient positive:

x^2 - 2x - 3 + b = 0.

This is a quadratic equation in $x$ whose solutions are the $x$ -coordinates of the intersection points.

Use the discriminant for the "exactly one real solution" condition

The number of real solutions of a quadratic $ax^2 + bx + c = 0$ depends on the discriminant $D = b^2 - 4ac$ :

If $D > 0$ , there are two distinct real solutions.
If $D = 0$ , there is exactly one real solution (a repeated root).
If $D < 0$ , there are no real solutions.

Here, $a = 1$ , $b_{\text{quad}} = -2$ , and $c = -3 + b$ . So the discriminant is

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

Simplify the discriminant and set it equal to 0

Simplify $D$ :

D = 4 - 4(-3 + b) \\[0.7em] = 4 + 12 - 4b \\[0.7em] = 16 - 4b.

For the system to have exactly one real solution, we need $D = 0$ :

16 - 4b = 0.

Now solve this equation for $b$ . Do not substitute answer choices yet; first find the exact value of $b$ that makes this true.

Solve for b and identify the correct choice

Solve

16 - 4b = 0.

Add $4b$ to both sides:

16 = 4b.

Divide both sides by $4$ :

b = 4.

So when $b = 4$ , the quadratic has exactly one real solution, meaning the line is tangent to the parabola and the system has exactly one real solution. The correct answer choice is C) 4.

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