Solve -2x² - 2x + 11 = 0
The solutions of -2x² - 2x + 11 = 0 are x = (-1 + √23)/2 or x = (-1 - √23)/2, that is x ≈ -2.8979158 and x ≈ 1.8979158.
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 = -2, c = 11.
- 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 ≈ -2.8979158 and x ≈ 1.8979158.
- 7
Check numerically: for x ≈ -2.8979158 the left side is 1.7763568e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -2x^2 - 2x + 11 = 0\ \Rightarrow\ x = \frac{-1 + \sqrt{23}}{2},\ \frac{-1 - \sqrt{23}}{2} \]
\[ -2x^2 - 2x + 11 = 0 \]
\[ D = (-2)^2 - 4\cdot(-2)\cdot11 = 4 + 88 = 92 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{92}}{-4} \]
\[ x = \frac{-1 + \sqrt{23}}{2}\quad\text{or}\quad x = \frac{-1 - \sqrt{23}}{2} \]The parabola y = -2x² - 2x + 11
| x | y |
|---|---|
| -3.5 | -6.5 |
| -2.5 | 3.5 |
| -1.5 | 9.5 |
| -0.5 | 11.5 |
| 0.5 | 9.5 |
| 1.5 | 3.5 |
| 2.5 | -6.5 |
Questions
What are the solutions of -2x² - 2x + 11 = 0?
The solutions of -2x² - 2x + 11 = 0 are x = (-1 + √23)/2 or x = (-1 - √23)/2, that is x ≈ -2.8979158 and x ≈ 1.8979158.
Does -2x² - 2x + 11 = 0 factor?
Not over the integers, because the discriminant 92 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -2x² - 2x + 11?
(-1/2, 23/2). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = -1, product = c/a = -11/2.