Solve by Element List Regression

No-Real-Solution Quadratic on the SAT: Desmos Walkthrough

Graph the answer choices and pick the parabola that never touches the

x

-axis (no gray intercept dot), since that one has no real solutions.

This lesson uses sliders and direct evaluation in Desmos: put the unknown constant on a slider or a single expression line and let the calculator find the value instead of solving algebra by hand.

Take this lesson interactively

A real SAT-style question

Which quadratic equation has no real solutions?

$x^{2} + 14 x - 49 = 0$

$x^{2} - 14 x + 49 = 0$

$5 x^{2} - 14 x - 49 = 0$

$5 x^{2} - 14 x + 49 = 0$

How to solve it in Desmos, step by step

The idea

Graph all four quadratics and find the one parabola that never touches the x-axis. A parabola with no x-intercept (no gray point on the x-axis) has no real solutions.

1
Step 1
Type this to graph choice A. Desmos marks where it crosses the x-axis with gray dots, so it has real solutions.
Type in Desmos
$y=x^{2}+14x-49$
2
Step 2
Type this to graph choice B. It touches the x-axis at exactly one gray dot, so it has one real solution.
Type in Desmos
$y=x^{2}-14x+49$
3
Step 3
Type this to graph choice C. It crosses the x-axis at two gray dots, so it has real solutions.
Type in Desmos
$y=5x^{2}-14x-49$
4
Step 4
Type this to graph choice D. This parabola sits entirely above the x-axis with no gray dot on the x-axis.
Type in Desmos
$y=5x^{2}-14x+49$
5
Step 5
Choice D is the only curve that never touches the x-axis, so it has no real solutions. The answer is D.

The answer is D.

$5 x^{2} - 14 x + 49 = 0$

Now you try one

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

Which quadratic equation has no real solutions?

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