One formula for all of them
Some equations have no pair of whole numbers that splits them. The formula still answers them.
x = (−b ± √(b² − 4ac)) / 2a. Step 1 of 5
x = (−b ± √(b² − 4ac)) / 2a.aandbcome from the standard form.csits under the root, withaagain.b² − 4acis the number under the root.±gives two roots from one line.
The rule
Take ax² + bx + c = 0, with a ≠ 0. When b² − 4ac is not below 0, the roots are x = (−b ± √(b² − 4ac)) / 2a.
Where it comes from
- Divide by
a:x² + (b/a)x + c/a = 0. - Move
c/a:x² + (b/a)x = −c/a. - Add
(b/2a)²to both sides. The left side is nowx² + (b/a)x + (b/2a)², which is(x + b/2a)². - The right side is
−c/a + b²/4a² = (b² − 4ac)/4a². - Take the square root:
x + b/2a = ±√(b² − 4ac)/2a. - Subtract
b/2afrom both sides. This gives the formula.
| Equation | a, b, c | b² − 4ac | Roots |
|---|---|---|---|
x² − 2x − 3 = 0 | 1, −2, −3 | 4 + 12 = 16 | (2 ± 4)/2, so 3 and −1 |
3x² − 5x + 2 = 0 | 3, −5, 2 | 25 − 24 = 1 | (5 ± 1)/6, so 1 and 2/3 |
2x² + 3x − 3 = 0 | 2, 3, −3 | 9 + 24 = 33 | (−3 ± √33)/4 |
Keep √33 as it is. A decimal is close, not exact.
Watch out!
For x² − 2x − 3, b is −2. Its square is 4, not −4. Write (−2) × (−2) to keep the sign right.
Now try it. Slide a, b and c. Pick the number that completes the square. The lab then shows each line with your numbers.
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