Solve by Graphing

Equations with Exactly One Solution on the SAT: Desmos Walkthrough

Find the constant that makes an equation have exactly one solution by graphing both sides in Desmos and sliding the unknown until the line is tangent to the parabola (touches at a single point).

This lesson solves by graphing: turn the equation into curves, then read intersections, vertices, or tangency straight off the Desmos graph.

Take this lesson interactively

A real SAT-style question

$- 9 x^{2} + 30 x + c = 0$

In the given equation, $c$ is a constant. The equation has exactly one solution. What is the value of $c$ ?

$3$

$0$

$negative 25$

$negative 53$

How to solve it in Desmos, step by step

The idea

Exactly one solution means the parabola just touches the x-axis at a single point. Add a slider for c and drag it until that happens. The answer is below the default slider range, so widen the slider first.

1
Step 1
Type this line. Desmos graphs the parabola and offers to add a slider for c. Click the button that says add slider for c.
Type in Desmos
$y=-9x^{2}+30x+c$
2
Step 2
On the c slider, click the lower bound number on the left and change it to negative 30, so the slider can reach low enough.
3
Step 3
Drag the c slider and watch the gray dots where the parabola meets the x-axis. Two dots means two solutions, no dots means none. Stop when there is exactly one dot, where the curve just touches the x-axis. The slider reads negative 25. The answer is C, negative 25.

The answer is C.

$negative 25$

Now you try one

Same question type, different numbers. Use the exact Desmos moves from the walkthrough above.

-4x^2 + 24x + c = 0

In the given equation,

c

is a constant. The equation has exactly one solution. What is the value of

c

Practice this with a live Desmos calculator

The interactive version of this lesson builds the Desmos graph up step by step as you click through, with a live calculator you can type into and your progress saved.

Open the interactive lesson

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