Quadratic Relations of the Form y = ax² + bx + c · Quadratic Expressions and Equations

Solving Quadratic Equations: Square Roots, Graphing and Choosing a Method

3.8About 65 minutes

Learning goals

I can solve quadratic equations by taking square roots, including equations in vertex form.
I can solve quadratic equations by graphing, using zeros or intersection points.
I can choose an efficient method (square roots, factoring, formula or graphing) and justify it.
I can interpret roots in a real context and reject those that do not make sense.

Words to know

square root property
If x² = k and k > 0, then x = √k or x = −√k.

point of intersection
A point that lies on two graphs. Its x-coordinate solves the equation formed by setting the two expressions equal.

real roots
Roots that are real numbers, so they appear as x-intercepts on the graph.

reject a root
Decide a correct root of the equation does not fit the situation, such as a negative time.

A toolbox, not a single tool

You now know four ways to solve a quadratic equation:

Square roots, for equations like x² = 49 or 2(x − 3)² − 8 = 0.
Factoring, when the expression factors nicely.
The quadratic formula, which always works.
Graphing, with technology, which gives decimal answers and a picture.

A carpenter could drive every screw with a hammer, but it would not go well. Choosing the right tool makes the job quick and reliable. This lesson is about choosing.

Solving by taking square roots

If an equation has the form (something)² = k, you can undo the square directly. Remember that both a positive and a negative number square to k:

If x² = 49, then x = 7 or x = −7 (written x = ±7).

This is the fastest method for equations in vertex form, because there is no x-term outside the bracket.

Worked example

Example 1: An equation in vertex form

Solve 2(x − 3)² − 8 = 0.

Isolate the square: 2(x − 3)² = 8, so (x − 3)² = 4.
Take the square root of both sides (±): x − 3 = 2 or x − 3 = −2.
x = 5 or x = 1.
Check x = 5: 2(2)² − 8 = 8 − 8 = 0 ✓. Check x = 1: 2(−2)² − 8 = 0 ✓.

x = 5 or x = 1. These are also the zeros of y = 2(x − 3)² − 8.

Worked example

Example 2: Two ways for the same equation

Solve 3x² = 48.

Square roots: x² = 16, so x = ±4.
Or factoring: 3x² − 48 = 0, 3(x² − 16) = 0, 3(x − 4)(x + 4) = 0, x = ±4. Same answer.
Do not forget the negative root.

x = 4 or x = −4

Solving by graphing

There are two ways to use a graph:

Zeros: rearrange to ax² + bx + c = 0, graph y = ax² + bx + c, and read the x-intercepts.
Intersections: graph each side of the equation as its own relation, and read the x-coordinates of the points where they meet.

Graphing technology gives accurate decimals. A hand-drawn graph gives estimates, which is still useful for checking.

Solving x² + 3x = 2x + 6 by graphing each side. The curves meet at (−3, 0) and (2, 10), so the solutions are the x-coordinates: x = −3 and x = 2.

Worked example

Example 3: Graph each side, then check algebraically

Solve x² + 3x = 2x + 6 by graphing, then confirm by factoring.

Graph y = x² + 3x and y = 2x + 6 (see above). They intersect at (−3, 0) and (2, 10).
The solutions are the x-coordinates: x = −3 and x = 2.
Algebraically: x² + 3x − 2x − 6 = 0, so x² + x − 6 = 0 and (x + 3)(x − 2) = 0.
x = −3 or x = 2 ✓. Both methods agree.

x = −3 or x = 2

Worked example

Example 4: Graphing a real model

Solve −4.9t² + 50t + 1.5 = 0 by graphing h = −4.9t² + 50t + 1.5 with technology, and interpret.

Graph y = −4.9x² + 50x + 1.5. Use a window that shows both zeros, for example x from −1 to 11.
The zeros are at about x = −0.03 and x = 10.23.
The equation has two roots: t ≈ −0.03 and t ≈ 10.23.
In the context of a kicked ball, t ≥ 0, so only t ≈ 10.23 s is meaningful. The negative root would be a time before the kick.

t ≈ 10.23 (the root t ≈ −0.03 is rejected in context).

A quick guide to choosing a method. Any method that works is correct, but the right one saves time and reduces errors.

Watch out

x² = 49 has two roots, ±7. Writing only 7 loses half the answer.
To use square roots, the squared part must be alone on one side first.
When graphing each side, the answers are the x-coordinates of the intersection points, not the y-coordinates.
In a context, a root can be mathematically correct but impossible (negative time or length). Say which roots you reject and why.

Hands-on activity

Method race

You need: a family member, two pencils, paper, a calculator, and a graphing tool if a grown-up says you may use one.

Copy these four equations onto a sheet: (a) x² − 9x + 20 = 0, (b) (x + 1)² = 36, (c) 2x² + x − 4 = 0, (d) x² + 2x = 8.
Before solving, write which method you think is fastest for each, and why.
Race your family member: you choose your methods; they must use the quadratic formula for all four (with the calculator).
Compare times and answers. Check one answer with a graph if you can.

Model answers: (a) factor: (x − 4)(x − 5) = 0, x = 4 or 5. (b) square roots: x + 1 = ±6, x = 5 or −7. (c) formula: x = (−1 ± √33) ÷ 4, x ≈ 1.19 or −1.69. (d) rearrange and factor: (x + 4)(x − 2) = 0, x = −4 or 2.

Just for fun

Mathematicians have been solving quadratic equations for nearly 4000 years. Clay tablets from ancient Babylon show problems about the sides of rectangles that lead to quadratic equations, solved step by step in a way that works like completing the square. No graphing calculators, and no excuses about a dead battery.

Your turn

Solve x² = 81.

Solve (x + 2)² = 25.

Solve 3(x − 1)² − 12 = 0.

Solve x² − 11x + 30 = 0. Which method did you use?

Solve 2x² + 3x − 4 = 0 to two decimal places.

Solve x² + 2x = 3x + 12.

Use graphing technology (or the formula) to solve 0.2x² − x − 3 = 0 to two decimal places.

More practice: warm-up

Solve x² = 49.

Solve x² − 36 = 0.

Solve (x − 4)² = 0.

Which method would you use for x² + 7x + 10 = 0? Solve it.

Which method would you use for 3x² + 5x − 1 = 0? Solve it to two decimal places.

More practice: on your own

Solve 4(x + 3)² = 36.

Solve 2x² − 3 = 5x.

Solve x² + 4x − 1 = 0 to two decimal places.

Solve (2x − 1)(x + 3) = 5 to two decimal places.

Find the points where y = x² − 4 and y = 3x intersect.

Challenge

Find the zeros of y = −(x − 2)² + 9 without expanding.

Solve x⁴ − 13x² + 36 = 0.

Try to solve x² + 9 = 0 by square roots, by the formula and by graphing. What happens each time?

Check your understanding

Quiz

Solve (x − 5)² = 16.

To solve x² + 3x = 2x + 6 by graphing, you can graph y = x² + 3x and y = 2x + 6 and then find:

Which equation is best solved by factoring?

Solve 5x² = 80.

What are the roots of 2(x + 1)² − 18 = 0?

For a kicked ball, −4.9t² + 50t + 1.5 = 0 has roots t ≈ −0.03 and t ≈ 10.23. Which answers the question "When does the ball land?"

For parents and teachers

Big idea: Expectation 3.8 asks students to solve quadratic equations that have real roots using a variety of methods: factoring, the quadratic formula and graphing. This lesson adds solving by square roots (useful for vertex form) and focuses on choosing a sensible method and checking with a second one.

How to help: Before your teen starts a question, ask "Which method, and why?" Any correct method earns full marks, but a quick choice saves time on tests. Encourage checking one answer with a graph.

Common difficulties:

Forgetting the ± when taking square roots.
Setting each bracket equal to a non-zero number, as in (2x − 1)(x + 3) = 5.
Reading y-coordinates instead of x-coordinates from intersection points.
Keeping roots that make no sense in context.