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

The Quadratic Formula: Where It Comes From and How to Use It

3.7 3.8About 75 minutes

Learning goals

I can follow the development of the quadratic formula by completing the square, and connect each step to a numerical example.
I can identify a, b and c after writing an equation as ax² + bx + c = 0.
I can solve quadratic equations with the quadratic formula, giving exact and approximate roots.
I can verify roots by substitution or with a graph.

Words to know

quadratic formula
x = (−b ± √(b² − 4ac)) ÷ 2a. It gives the roots of ax² + bx + c = 0.

± (plus or minus)
A sign that means "do the calculation twice, once adding and once subtracting."

exact answer
An answer left with square roots or fractions, such as (−1 ± √17) ÷ 2.

approximate answer
A rounded decimal answer, such as x ≈ 1.56.

algebraic development
A step-by-step argument that shows where a formula comes from.

When factoring fails

Try to solve x² + x − 4 = 0 by factoring. You need two integers with product −4 and sum 1. The pairs are 1 and −4 (sum −3), −1 and 4 (sum 3), and 2 and −2 (sum 0). None works.

But the graph of y = x² + x − 4 clearly crosses the x-axis twice, so the equation does have two roots. They just are not nice numbers. We need a method that works for every quadratic equation. Completing the square is that method, and if we do it once for the general equation ax² + bx + c = 0, we get a formula that saves us from ever doing it again.

y = x² + x − 4 has two zeros, but they are not integers, so factoring over the integers cannot find them. The quadratic formula can.

Where the formula comes from

The table below completes the square on a numerical example (left) and on the general equation (right) side by side. Read across each row: the same step is done to both. You do not have to reproduce the general version yourself, but you should be able to follow it and connect each step to the example.

Step	Example: 2x² + 8x + 3 = 0 (a = 2, b = 8, c = 3)	General: ax² + bx + c = 0
1. Divide every term by a	x² + 4x + 3/2 = 0	x² + (b/a)x + c/a = 0
2. Move the constant to the right side	x² + 4x = −3/2	x² + (b/a)x = −c/a
3. Add (half the x-coefficient)² to both sides	x² + 4x + 4 = 4 − 3/2	x² + (b/a)x + b²/(4a²) = b²/(4a²) − c/a
4. Factor the left side; use a common denominator on the right	(x + 2)² = 5/2	(x + b/(2a))² = (b² − 4ac)/(4a²)
5. Take the square root of both sides (±)	x + 2 = ±√(5/2)	x + b/(2a) = ±√(b² − 4ac) / (2a)
6. Isolate x	x = −2 ± √(5/2)	x = (−b ± √(b² − 4ac)) / (2a)

Developing the quadratic formula by completing the square. In step 3 the example adds (4 ÷ 2)² = 4; the general case adds (b/(2a))² = b²/(4a²).

Checking the connection

Finish the example with a calculator: √(5/2) ≈ 1.581, so x ≈ −2 + 1.581 = −0.42 or x ≈ −2 − 1.581 = −3.58.

Now put a = 2, b = 8, c = 3 straight into the finished formula:

x = (−8 ± √(64 − 24)) ÷ 4 = (−8 ± √40) ÷ 4 ≈ (−8 ± 6.325) ÷ 4,

which gives x ≈ −0.42 or x ≈ −3.58. The same answers, in one line instead of six.

The quadratic formula. It solves any quadratic equation once it is written as ax² + bx + c = 0. Many students memorize it as a short song (see below).

Worked example

Example 1: Solve x² + x − 4 = 0

Solve x² + x − 4 = 0 using the quadratic formula. Give exact answers and answers to two decimal places, and verify graphically.

Identify a = 1, b = 1, c = −4.
Substitute: x = (−1 ± √(1² − 4(1)(−4))) ÷ (2 × 1).
Simplify inside the root: 1 − (−16) = 1 + 16 = 17. So x = (−1 ± √17) ÷ 2.
With +: (−1 + 4.123) ÷ 2 ≈ 1.56. With −: (−1 − 4.123) ÷ 2 ≈ −2.56.
Verify graphically: the graph of y = x² + x − 4 above crosses the x-axis at about −2.56 and 1.56 ✓.

Exact: x = (−1 ± √17) ÷ 2. Approximate: x ≈ 1.56 or x ≈ −2.56.

Worked example

Example 2: The formula also works when factoring would

Solve 3x² − 5x − 2 = 0 with the formula, then check by factoring.

a = 3, b = −5, c = −2. Note −b = +5.
b² − 4ac = (−5)² − 4(3)(−2) = 25 + 24 = 49.
x = (5 ± √49) ÷ 6 = (5 ± 7) ÷ 6.
x = 12 ÷ 6 = 2 or x = −2 ÷ 6 = −1/3.
Check by factoring: (3x + 1)(x − 2) = 3x² − 6x + x − 2 = 3x² − 5x − 2 ✓.

x = 2 or x = −1/3. When the number under the root is a perfect square, the equation could also be factored.

Using the formula without mistakes

First write the equation as ax² + bx + c = 0. Then list a, b and c with their signs.
Use brackets when you substitute negatives: b² = (−5)² = 25, not −25.
−b means "the opposite of b": if b = −5, then −b = 5.
The whole numerator is divided by 2a, not just the square root.
On a calculator, work out √(b² − 4ac) first, then do the + and − versions separately.

Hands-on activity

Formula practice with a calculator

You need: a calculator (a phone calculator works), paper and a pencil, and a family member.

Ask a family member to choose three integers for a, b and c, with a positive and c negative (then −4ac is positive, so b² − 4ac is positive, which guarantees two roots).
Write the equation ax² + bx + c = 0 and solve it with the formula. Round to two decimal places.
Check each root by substituting it back into ax² + bx + c. Because of rounding, you should get a number very close to 0 (for example, 0.01 or −0.02).
Teach your family member the "song" below and see who can say the formula fastest.

Model answer: for a = 2, b = 3, c = −5: b² − 4ac = 9 + 40 = 49, so x = (−3 ± 7) ÷ 4, giving x = 1 or x = −2.5. Check: 2(1) + 3 − 5 = 0 ✓ and 2(6.25) − 7.5 − 5 = 0 ✓.

Just for fun

Many students remember the quadratic formula by singing it to the tune of "Pop Goes the Weasel": "x equals negative b, plus or minus the square root, of b squared minus four a c, all over two a!" Try not to sing it out loud during a test.

Your turn

State a, b and c for 2x² − 3x − 7 = 0 and for 5 = x² − 4x.

Solve x² + 5x + 2 = 0. Give answers to two decimal places.

Solve 2x² − 7x + 3 = 0 using the formula.

Solve x² − 6x + 4 = 0. Give exact and approximate answers.

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

Solve x² = 3x + 1 to two decimal places.

Solve −2x² + 4x + 5 = 0 to two decimal places.

More practice: warm-up

Write the quadratic formula.

State a, b and c for 4x² + x = 9.

Calculate b² − 4ac for x² + 3x − 10 = 0.

Solve x² + 3x − 10 = 0 with the formula.

Evaluate (−4 ± √36) ÷ 2.

More practice: on your own

Solve 5x² − 2x − 1 = 0 to two decimal places.

Solve x(x + 4) = 7 to two decimal places.

Solve 0.5x² − 2x − 3 = 0 to two decimal places.

A rectangular garden is 2 m longer than it is wide, with area 20 m². Find its dimensions to two decimal places.

A student solved x² − 4x − 3 = 0 by writing x = (−4 ± √(16 + 12)) ÷ 2. Find the error and give the correct roots.

Challenge

Show that the two roots from the quadratic formula always add up to −b ÷ a. Test it on 2x² − 7x + 3 = 0.

A ball is thrown upward from 1.5 m. Its height is h = −4.9t² + 12t + 1.5. When does it hit the ground? Answer to two decimal places.

Use the formula to explain why x² − 6x + 9 = 0 has only one root.

Check your understanding

Quiz

Which is the quadratic formula for ax² + bx + c = 0?

For 3x² − x − 5 = 0, what is b?

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

What is the first step to solve 2x² = 5x + 1 with the formula?

The quadratic formula is developed by applying which method to ax² + bx + c = 0?

What are the roots of 2x² + 8x + 3 = 0, to two decimal places?

For parents and teachers

Big idea: Expectation 3.7 asks students to explore the algebraic development of the quadratic formula by following a demonstration and connecting the steps to a numerical example; reproducing the general derivation is not required. Expectation 3.8 then uses the formula to solve equations such as the curriculum sample x² + x − 4 = 0, verified graphically.

How to help: Read the side-by-side table together, one row at a time, asking "What did they do to both sides here?" When your teen uses the formula, check that they wrote a, b and c with signs before substituting.

Common difficulties:

Forgetting to rearrange to = 0 first.
Sign errors with −b and with b² when b is negative.
Dividing only the square root by 2a instead of the whole numerator.
Rounding too early. Keep the calculator value until the end.