Desmos Power Moves I: Solve by Graphing

Type an equation, system, or inequality exactly as written, and let Desmos show you every solution.

28 min
Desmos way

Ready when you are

7 steps · about 28 minutes

Start with the warm-up.

Start

Step 1 of 7Warm-upDone

Warm-up

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Key terms

Solution: A value that makes an equation true. On a graph, it is where the two sides meet.
Point of interest: A gray dot Desmos marks at an intercept, a vertex, or an intersection. Click it to see its coordinates.
System of equations: Two or more equations that must be true at the same time. Its solution is where the graphs cross.
Strict inequality: An inequality that uses $<$ or $>$ . Its boundary is not included, so Desmos draws it dashed.

Step 2 of 7Try it firstDone

Try it first

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Take a shot before we teach anything. There’s no penalty, and Desmos is fair game if you already know a move.

Step 3 of 7LearnDone

The big idea

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Let the graph do the solving

An equation asks one question: where do the left side and the right side agree? Picture two hiking trails. The only places two hikers can meet are where the trails cross. A graph shows you those spots.

Take $x^2 - 2x = 3$ . Graph $y = x^2 - 2x$ and $y = 3$ . They cross twice, at $x = -1$ and $x = 3$ , so those are the two solutions.

y = x^2 - 2x

y = 3

Desmos can skip the splitting. Type the equation exactly as written. There’s no $y$ , so Desmos draws a vertical line at every $x$ that works.

Explore in Desmos

Watch the solutions appear

Type the equation as-is, then click each vertical line. How many solutions are there, and what are they?

The calculator is live: change anything and see what happens.

Watch the solutions appear

Degree mode

Loading Desmos…

Key idea

Two ways to solve a one-variable equation in Desmos

Type it as-is: for equations Desmos can graph implicitly, each vertical solution line marks a valid input. Or split it: graph $y =$ the left side and $y =$ the right side, then click each intersection. The $x$ -coordinates are the solutions. Widen the viewing window and check the original domain before deciding you found them all.

Systems, zeros, and vertices

For a system, type both equations as written. Don’t solve for $y$ first. Click where the graphs cross: that point is the solution.

Crossing lines: exactly one solution.
Parallel lines: no solution.
Same line: infinitely many. You’ll see just one line, because the second sits right on top of the first.

For a single graph like $y = x^2 - 4x - 5$ , Desmos marks gray points of interest. Click them. The $x$ -intercepts $(-1, 0)$ and $(5, 0)$ are the zeros, and the lowest point, $(2, -9)$ , is the vertex.

Counting solutions of a linear system

\text{crossing} \rightarrow 1 \qquad \text{parallel} \rightarrow 0 \qquad \text{same line} \rightarrow \text{infinitely many}

Same slope means parallel or the same line. Then compare the $y$ -intercepts.

Inequalities and absolute value

Type an inequality and Desmos shades the points that work. Read the boundary: dashed means the edge is not included ( $<$ or $>$ ), and solid means it is ( $\le$ or $\ge$ ). This gives you a visual check on the direction of an inequality after dividing by a negative.

For absolute value, press the | key or type abs(.

Explore in Desmos

Read the shading

Where is the shading? Are its edges solid or dashed? Write the solution set.

The calculator is live: change anything and see what happens.

Read the shading

Degree mode

Loading Desmos…

Trap

The half-the-answer trap

$(x + 4)^2 = 9$ means $x + 4 = 3$ or $x + 4 = -3$ , and $|x - 1| = 6$ has two cases too. Forget one and you have half the answer. Count the vertical lines in Desmos, check whether another solution could be outside the window, then reread the ask: one solution, the greater one, or their sum?

When Desmos gives decimals

Desmos shows decimals like $5.236$ , but ACT choices are often exact, like $3 + \sqrt{5}$ . Type each choice, such as 3+sqrt(5), and compare more digits if two values look close. A rounded display is useful for separating choices; it is not proof that two exact expressions are equal.

Reach for algebra when a one-step problem is faster in your head, or when the choices contain a letter, like $3k + 1$ . A legal test value can eliminate a symbolic choice that fails; one matching value cannot prove an identity. Use a second value to separate remaining choices, or verify algebraically.

Quadratic formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

The $\pm$ is why a quadratic can have two solutions. If $b^2 - 4ac$ is negative, the parabola never touches the $x$ -axis, so there are no real solutions.

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Worked examples

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Skip check · 2 questions

Already know this? Get both right on the first try and you can skip the worked examples.

Worked example

Example 1: Type it as-is

Problem

What are all the solutions of

x^2 + x = 12

Type the equation exactly as written: x^2+x=12. No rearranging.

The equation has no $y$ , so Desmos draws vertical lines. How many should you expect?

Predict the next move

Click each line. Desmos labels them

x = -4

and

x = 3

Check one: $(-4)^2 + (-4) = 16 - 4 = 12$ . The solutions are $-4$ and $3$ . Why does a vertical line mean “solution”?

Why does this step work?

Worked example

Example 2: Read the ask, then match the choices

Problem

In the standard

(x,y)

coordinate plane, the graphs of

y = x^2 - 2x - 3

and

y = x + 1

intersect at two points. In simplest radical form, what is the distance between those two points?

Type y=x^2-2x-3 on line 1 and y=x+1 on line 2.

Click both crossings: $(-1, 0)$ and $(4, 5)$ . What’s next?

Predict the next move

On line 3, type sqrt((4-(-1))^2+(5-0)^2). Desmos prints

7.0710678

The ask wants an exact radical, so type a candidate: 5sqrt(2). It also prints $7.0710678$ , so the distance is $5\sqrt{2}$ .

Why does this step work?

Faded example · you finish it

Example 3: You finish it

Problem

Solve

3 - 2x \ge 11

Subtract

3

from both sides:

-2x \ge 8

Divide both sides by $-2$ . The result is

In Desmos, the boundary at $x = -4$ looks like

Desmos way

The Desmos way (10 seconds)

Type 3-2x>=11. Desmos turns >= into

\ge

as you type.

The shading sits to the left of a solid vertical line. Click the line:

x = -4

Shading left of a solid edge means

x \le -4

. There’s no sign flip to remember, because Desmos never divided by anything.

The Desmos way (10 seconds)

Degree mode

Loading Desmos…

Step 5 of 7PracticeDone

Practice

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Speed drills

Race the clock. Each drill is one Desmos move: type, click, read, answer.

Step 6 of 7Spot the mistakeDone

Spot the mistake

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Spot the mistake

Where did it go wrong?

One line has an error. Click the line where it first goes wrong.

Desmos would have caught it: 2(x-1)^2=50 draws two vertical lines, at $x = -4$ and $x = 6$ .

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Exit ticket

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