Feed a, b, c to the machine
One formula gives the roots of any quadratic equation.
One recipe for all?
Splitting the middle term needs a good guess. Compare x² + 7x − 60 = 0 and 2x² + x − 300 = 0. Both have the same shape, ax² + bx + c = 0. Only the numbers a, b, c differ.
Could one recipe, using only a, b and c, give the roots of every quadratic equation?
What this lesson covers
The idea
The roots of ax² + bx + c = 0 are given by x = (−b ± √(b² – 4ac))/2a, provided b² – 4ac ≥ 0.
One recipe for all?
Splitting the middle term needs a good guess. Compare x² + 7x − 60 = 0 and 2x² + x − 300 = 0. Both have the same shape, ax² + bx + c = 0. Only the numbers a, b, c differ.
Could one recipe, using only a, b and c, give the roots of every quadratic equation?
- No: each equation needs its own trick
- Yes: a formula in a, b and c
- Only when a = 1
Turn the handle
Read a, b and c (with their signs) by tapping the right numbers. Then turn the handle: the numbers go into the formula, step by step. Try all four equations.
The quadratic formula
The roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / 2a, provided b² − 4ac ≥ 0.
The ± gives two roots: one with + and one with −. For x² + 7x − 60 = 0: b² − 4ac = 49 + 240 = 289 and √289 = 17, so x = (−7 ± 17)/2, which is 5 or −12.
Look at the two roots together: they add up to −b/a, and they multiply to c/a. This is the same as what you found for the zeroes of a quadratic polynomial in Chapter 2. And the roots sit equally far on either side of −b/2a.
Notes
The roots of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) / 2a, provided b² − 4ac ≥ 0.
Check yourself
For x² − 5x + 6 = 0, find b² − 4ac.
Answer: 1
b² − 4ac = (−5)² − 4 × 1 × 6 = 25 − 24 = 1.
With b² − 4ac = 1, the roots of x² − 5x + 6 = 0 are (5 ± 1)/2. What is the larger root?
Answer: 3
x = (5 + 1)/2 = 3 or x = (5 − 1)/2 = 2. Check: 9 − 15 + 6 = 0 ✓.
Use the formula on x² + 2x − 15 = 0. What is the positive root?
Answer: 3
x = (−2 + 8)/2 = 3 or x = (−2 − 8)/2 = −5. Check: 9 + 6 − 15 = 0 ✓.
For x² + 2x + 5 = 0, b² − 4ac = 4 − 20 = −16. What does the formula tell us?
- There are no real roots: √(−16) is not a real number — correct. Yes! The formula needs b² − 4ac ≥ 0. Here the machine stops.
- x = (−2 ± 4)/2, so x = 1 or −3. No real number squared is −16, so √(−16) is not 4.
- x = −1 only. There is no real number that is √(−16), so there is no root at all.