Solve -2x² - 11x + 12 = 0
The solutions of -2x² - 11x + 12 = 0 are x = (-11 + √217)/4 or x = (-11 - √217)/4, that is x ≈ -6.43273 and x ≈ 0.93272997.
Change a, b or c (whole numbers, a not 0) and the exact solutions follow.
How to work it out
- 1
Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -2, b = -11, c = 12.
- 2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
- 3
D is positive, so there are two different real solutions.
- 4
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
- 6
As decimals: x ≈ -6.43273 and x ≈ 0.93272997.
- 7
Check numerically: for x ≈ -6.43273 the left side is -1.4210855e-14 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -2x^2 - 11x + 12 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{217}}{4},\ \frac{-11 - \sqrt{217}}{4} \]
\[ -2x^2 - 11x + 12 = 0 \]
\[ D = (-11)^2 - 4\cdot(-2)\cdot12 = 121 + 96 = 217 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{217}}{-4} \]
\[ x = \frac{-11 + \sqrt{217}}{4}\quad\text{or}\quad x = \frac{-11 - \sqrt{217}}{4} \]The parabola y = -2x² - 11x + 12
| x | y |
|---|---|
| -5.75 | 9.125 |
| -4.75 | 19.125 |
| -3.75 | 25.125 |
| -2.75 | 27.125 |
| -1.75 | 25.125 |
| -0.75 | 19.125 |
| 0.25 | 9.125 |
Questions
What are the solutions of -2x² - 11x + 12 = 0?
The solutions of -2x² - 11x + 12 = 0 are x = (-11 + √217)/4 or x = (-11 - √217)/4, that is x ≈ -6.43273 and x ≈ 0.93272997.
Does -2x² - 11x + 12 = 0 factor?
Not over the integers, because the discriminant 217 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -2x² - 11x + 12?
(-11/4, 217/8). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -11/2, product = c/a = -6.