← Back to course
Algebra 1 9-12 / Week 10 / Tuesday
2/6
Week 10 · Quadratic Equations

Tuesday

Four ways to solve
// When is the height zero?
⏱ about 20 min

Tuesday: Four Ways to Solve

Wren pins four index cards to the Loft wall. "Factoring. Square roots. Completing the square. The quadratic formula."

"Four ways?" Comet says. "Factoring worked fine yesterday."

"Try x² + 6x + 2 = 0," Wren says. "Find two whole numbers that multiply to 2 and add to 6."

Comet tries for a minute. "There are none."

Nova dims her light to a hint. "What if the left side were a perfect square plus a number?"

"x² + 6x + 9 is (x + 3)²," Comet says slowly. "So x² + 6x + 2 is (x + 3)² minus 7."

"Then (x + 3)² = 7, and x + 3 is plus or minus the square root of 7," Wren says.

"Completing the square," Comet says. "Okay, I see why we need four cards."

Way one: factoring

When the quadratic factors over whole numbers, factor and use the zero-product rule.

For example, x² - 7x + 12 = (x - 4)(x - 3). So x = 3 or x = 4.

Factoring is fastest when it works. It does not work for every quadratic.

Way two: square roots

When the equation has a square alone on one side, take the square root of both sides. Remember both signs.

For example, x² = 49 gives x = 7 or x = -7.

And (x - 2)² = 9 gives x - 2 = 3 or x - 2 = -3. So x = -1 or x = 5.

Way three: completing the square

  1. Start with x² + 6x + 2 = 0 and move the number: x² + 6x = -2.
  2. Take half of 6, which is 3, and square it: 9. Add 9 to both sides: x² + 6x + 9 = 7.
  3. The left side is now a perfect square: (x + 3)² = 7.
  4. Take square roots: x + 3 = √7 or x + 3 = -√7.
  5. So x = -3 - √7 or x = -3 + √7, about -5.65 and -0.35.

Way four: the quadratic formula

For any ax² + bx + c = 0, the solutions are x = (-b ± √(b² - 4ac)) ÷ 2a. It comes from completing the square on the general equation.

For 2x² + 3x - 5 = 0: a = 2, b = 3, c = -5. The root part is √(9 + 40) = √49 = 7.

So x = (-3 + 7) ÷ 4 = 1 or x = (-3 - 7) ÷ 4 = -2.5.

Compare the four ways

EquationBest wayWhySolutions
x² - 7x + 12 = 0factoring3 × 4 = 12 and 3 + 4 = 7x = 3 or 4
(x - 2)² = 9square rootsa square is already alonex = -1 or 5
x² + 6x + 2 = 0completing the squarea = 1 and b is evenx = -3 - √7 or -3 + √7
2x² + 3x - 5 = 0the formulaa is not 1 and it looks messyx = -2.5 or 1

Every way gives the same solutions. Wren's rule: pick the way that is quickest for the equation in front of you, then check.

WHICH WAY WOULD YOU PICK?
  • Read the question.
  • Tap your answer.
x² = 49. Which way is quickest?
x² - 7x + 12 = 0. Which way is quickest?
x² + 6x + 2 = 0 has no whole-number factors. Which way did the crew use?
COMPLETE THE SQUARE FOR X² + 6X + 2 = 0
  • ?Add 9 to both sides: x² + 6x + 9 = 7
  • ?Write the perfect square: (x + 3)² = 7
  • ?Move the number: x² + 6x = -2
  • ?Take square roots: x + 3 = ±√7
  • ?Solve: x = -3 ± √7
WHY THIS EXERCISECompleting the square turns any quadratic into a square-root problem. The formula is this method done once for all.
SOLVE FOUR WAYS
  • Read the question.
  • Tap your answer.
Solve x² - 7x + 12 = 0 by factoring. What are the solutions?
Solve (x - 2)² = 9 by taking square roots. What are the solutions?
Complete the square for x² + 4x + 1 = 0. What are the exact solutions?
Use the quadratic formula on 2x² + 3x - 5 = 0. What are the solutions?
StatementTrue or false?
x² = 49 has two solutions, 7 and -7.?
3 × 3 - 4 × 2 × (0 - 5) = 49?
Completing the square on x² + 6x adds 36.?
The quadratic formula works for every quadratic equation.?
(0 - 3 + 7) ÷ 4 = 1?
WHY THIS EXERCISEAll four ways agree because they are the same algebra in different clothes. Checking confirms it.
Try it
Write x² - 2x - 8 = 0 on a card. Solve it by factoring, then again with the quadratic formula.
Time each way with a stopwatch. Which was quicker for you, and did both agree?

Excellent. Tomorrow the rocket flies again in the Loft Lab and your stopwatch builds the equation.

← Monday