A. Quadratic Functions · Quadratic Expressions and EquationsCompleting the Square and Developing the Quadratic Formula
A1.6About 75 minutes
Learning goals- I can solve equations of the form (x + p)² = k by taking square roots.
- I can solve a quadratic equation by completing the square.
- I can follow the development of the quadratic formula and connect each step to a numerical example.
Words to know
square root property
If X² = k with k ≥ 0, then X = √k or X = −√k.
completing the square
Adding (b/2)² to x² + bx to make the perfect square (x + b/2)².
quadratic formula
x = (−b ± √(b² − 4ac)) / (2a), the roots of ax² + bx + c = 0.
exact value
An answer written with radicals or fractions, such as −2 + √5, rather than a rounded decimal.
computer algebra system (CAS)
Software that manipulates algebraic expressions and solves equations symbolically.
When factoring fails
Try to solve x² + 4x − 1 = 0 by factoring. You need two integers with product −1 and sum 4. The only pairs are 1 and −1 (sum 0). Factoring fails.
Yet the graph clearly crosses the x-axis twice, near −4.24 and 0.24. The roots exist; they just are not "nice" numbers. We need a method that works for every quadratic equation. That method is completing the square, and doing it once for the general equation ax² + bx + c = 0 produces the famous quadratic formula.
Tool 1: square roots
If an equation looks like (something)² = k with k ≥ 0, take the square root of both sides. Remember both signs, because 3² = 9 and (−3)² = 9.
(x + 2)² = 5 → x + 2 = ±√5 → x = −2 ± √5, so x ≈ 0.24 or x ≈ −4.24.
If k is negative, there is no real solution: no real number squared is negative.
Tool 2: completing the square
To use square roots on x² + 4x − 1 = 0, we need the left side to be a perfect square. Move the constant: x² + 4x = 1. Now add the number that completes the square. For x² + bx, that number is (b/2)²: half the coefficient of x, squared.
Worked exampleExample 1: Solve by completing the square
- Move the constant: x² + 4x = 1.
- Half of 4 is 2, and 2² = 4. Add 4 to both sides: x² + 4x + 4 = 5.
- Write the left side as a square: (x + 2)² = 5.
- Square roots: x + 2 = ±√5.
- x = −2 + √5 ≈ 0.236 or x = −2 − √5 ≈ −4.236. These match the graph.
x = −2 ± √5 (x ≈ 0.24 or x ≈ −4.24)
Developing the quadratic formula
Now do exactly the same steps to the general equation ax² + bx + c = 0 (with a ≠ 0). The table puts a numerical example beside the general case so you can connect each step. You are not expected to reproduce the general version from memory, but you should be able to follow it and explain each step with the example.
| Step | Example: 2x² + 6x − 3 = 0 | General: ax² + bx + c = 0 |
|---|
| 1. Move the constant | 2x² + 6x = 3 | ax² + bx = −c |
| 2. Divide by a | x² + 3x = 3/2 | x² + (b/a)x = −c/a |
| 3. Add (half the x-coefficient)² | x² + 3x + 9/4 = 3/2 + 9/4 | x² + (b/a)x + b²/(4a²) = −c/a + b²/(4a²) |
| 4. Write as a square; combine the right side | (x + 3/2)² = 15/4 | (x + b/(2a))² = (b² − 4ac)/(4a²) |
| 5. Take square roots | x + 3/2 = ±√15 / 2 | x + b/(2a) = ±√(b² − 4ac) / (2a) |
| 6. Isolate x | x = (−3 ± √15) / 2 | x = (−b ± √(b² − 4ac)) / (2a) |
The quadratic formula
For ax² + bx + c = 0 with a ≠ 0:
x = (−b ± √(b² − 4ac)) / (2a)
The ± gives two roots: one with + and one with −. The whole numerator, −b ± √(b² − 4ac), is divided by 2a.
Worked exampleExample 2: Connecting the formula to the example
Use the formula on 2x² + 6x − 3 = 0 and compare with step 6 of the table.
- a = 2, b = 6, c = −3.
- b² − 4ac = 36 − 4(2)(−3) = 36 + 24 = 60.
- x = (−6 ± √60) / 4. With a calculator, √60 ≈ 7.746.
- x ≈ (−6 + 7.746) / 4 ≈ 0.436 or x ≈ (−6 − 7.746) / 4 ≈ −3.436.
- From the table: (−3 ± √15) / 2, with √15 ≈ 3.873, gives 0.436 and −3.436. The same roots, because √60 = 2√15, so (−6 ± 2√15)/4 = (−3 ± √15)/2.
x ≈ 0.44 or x ≈ −3.44, by both methods
Following the development with technology
A computer algebra system (CAS), such as the free GeoGebra CAS app, can do the algebra for you. Type the general equation into a solve command (in GeoGebra CAS, something like Solve(a x^2 + b x + c = 0, x)), and it returns the quadratic formula. You can also type each step of the table into the CAS and watch it simplify, which is a good way to check that each step is legal. Graphing technology does the reverse check: graph y = 2x² + 6x − 3 and read the x-intercepts (about −3.44 and 0.44).
Just for funMany 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 it. It sticks.
Hands-on activityComplete a real square
You need: paper, a ruler, a pencil and scissors (take care with scissors).
- Cut a 10 cm by 10 cm square (call the side x, so this is x²).
- Cut two strips 10 cm by 3 cm (each is 3x). Together with the square these show x² + 6x.
- Place one strip on the right of the square and the other along the bottom. What shape is missing to make a bigger square?
- Cut that missing piece. What are its dimensions and area?
What to look for: the missing corner is 3 cm by 3 cm, area 9 = (6/2)². Adding it gives a 13 cm by 13 cm square: x² + 6x + 9 = (x + 3)². Check: 100 + 60 + 9 = 169 = 13².
Your turn
x − 3 = ±4: x = 7 or x = −1
2Solve (x + 1)² = 7. Give exact and approximate answers.
x = −1 ± √7, so x ≈ 1.65 or x ≈ −3.65
3What number completes the square x² + 10x + ___? Write the result as a square.
(10/2)² = 25: x² + 10x + 25 = (x + 5)²
4Solve x² + 6x − 2 = 0 by completing the square.
x² + 6x = 2; add 9: (x + 3)² = 11; x = −3 ± √11 ≈ 0.32 or −6.32
5Solve x² − 8x + 3 = 0 by completing the square.
x² − 8x = −3; add 16: (x − 4)² = 13; x = 4 ± √13 ≈ 7.61 or 0.39
6In the development of the formula, why do we divide by a in step 2?
So the x² coefficient is 1. Only then can x² + (b/a)x + (something) be written as a perfect square (x + p)².
7Use the quadratic formula on x² + 6x − 2 = 0 and compare with your answer above.
a = 1, b = 6, c = −2: x = (−6 ± √(36 + 8)) / 2 = (−6 ± √44) / 2. √44 = 2√11, so x = −3 ± √11. The same roots.
More practice: warm-up
x + 4 = ±3: x = −1 or x = −7
3Complete the square: x² − 12x + ___.
4Complete the square: x² + 3x + ___.
5Write the quadratic formula.
x = (−b ± √(b² − 4ac)) / (2a)
More practice: on your own
1Solve 2(x − 1)² − 18 = 0.
2(x − 1)² = 18, (x − 1)² = 9, x − 1 = ±3: x = 4 or x = −2
2Solve x² + 2x − 5 = 0 by completing the square.
x² + 2x = 5; add 1: (x + 1)² = 6; x = −1 ± √6 ≈ 1.45 or −3.45
3Solve 2x² + 8x − 1 = 0 by completing the square, then check with the formula.
Divide by 2: x² + 4x = 1/2; add 4: (x + 2)² = 4.5; x = −2 ± √4.5 ≈ 0.12 or −4.12. Formula: (−8 ± √(64 + 8)) / 4 = (−8 ± 8.485) / 4 ≈ 0.12 or −4.12 ✓.
4In the general development, what number is added in step 3, and why?
b²/(4a²). The coefficient of x is b/a; half of it is b/(2a); squared, it is b²/(4a²). Adding it makes the left side the perfect square (x + b/(2a))².
2x + 1 = ±5: 2x = 4 or 2x = −6, so x = 2 or x = −3
Challenge
1Solve 3x² − 6x − 2 = 0 by completing the square, and check with the formula.
Divide by 3: x² − 2x = 2/3; add 1: (x − 1)² = 5/3; x = 1 ± √(5/3) ≈ 2.29 or −0.29. Formula: (6 ± √(36 + 24)) / 6 = (6 ± 7.746) / 6 ≈ 2.29 or −0.29 ✓.
2Explain why (x + 3)² = −4 has no real solutions. Which part of the quadratic formula would show this?
A square of a real number is never negative, so (x + 3)² cannot equal −4. In the formula the same thing happens when b² − 4ac < 0: the square root of a negative number is not a real number. (Expanded, this equation is x² + 6x + 13 = 0, and b² − 4ac = 36 − 52 = −16.)
3Jordan writes the formula as x = −b ± √(b² − 4ac) / 2a. Use 2x² + 6x − 3 = 0 to show why this is wrong.
Jordan divides only the square root by 2a. With + : −6 + √60/4 ≈ −6 + 1.94 = −4.06, but 2(−4.06)² + 6(−4.06) − 3 ≈ 33.0 − 24.4 − 3 ≈ 5.6 ≠ 0. The correct x = (−6 + √60)/4 ≈ 0.44 makes the left side 0. The whole numerator must be divided by 2a.
Quiz
2. What number completes the square x² + 14x + ___?
3. In the development of the quadratic formula, which method is applied to ax² + bx + c = 0?
4. Which is the quadratic formula?
5. Why does x² = −9 have no real solutions?
6. (x + 3)² = 8. The exact solutions are:
Big idea (A1.6, first part): students explore where the quadratic formula comes from. The curriculum says they should follow the algebraic development and connect it to a numerical example, not reproduce the general derivation themselves.
How to help: cover the "General" column of the table and ask your teen to explain each step of the numerical example. Then uncover it and ask, "Which general step matches this one?" If you know the Pop Goes the Weasel song, sing it together.
Common difficulties:
- Forgetting the ± when taking square roots (losing one root).
- Adding (b/2)² to one side only.
- Forgetting to divide by a before completing the square.
- Dividing only the square root by 2a instead of the whole numerator.