Solve 5x² - 9x - 11 = 0
The solutions of 5x² - 9x - 11 = 0 are x = (9 + √301)/10 or x = (9 - √301)/10, that is x ≈ 2.6349352 and x ≈ -0.83493516.
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 = 5, b = -9, 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 ≈ 2.6349352 and x ≈ -0.83493516.
- 7
Check numerically: for x ≈ 2.6349352 the left side is -7.1054274e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 5x^2 - 9x - 11 = 0\ \Rightarrow\ x = \frac{9 + \sqrt{301}}{10},\ \frac{9 - \sqrt{301}}{10} \]
\[ 5x^2 - 9x - 11 = 0 \]
\[ D = (-9)^2 - 4\cdot5\cdot(-11) = 81 + 220 = 301 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{301}}{10} \]
\[ x = \frac{9 + \sqrt{301}}{10}\quad\text{or}\quad x = \frac{9 - \sqrt{301}}{10} \]The parabola y = 5x² - 9x - 11
| x | y |
|---|---|
| -2.1 | 29.95 |
| -1.1 | 4.95 |
| -0.1 | -10.05 |
| 0.9 | -15.05 |
| 1.9 | -10.05 |
| 2.9 | 4.95 |
| 3.9 | 29.95 |
Questions
What are the solutions of 5x² - 9x - 11 = 0?
The solutions of 5x² - 9x - 11 = 0 are x = (9 + √301)/10 or x = (9 - √301)/10, that is x ≈ 2.6349352 and x ≈ -0.83493516.
Does 5x² - 9x - 11 = 0 factor?
Not over the integers, because the discriminant 301 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 5x² - 9x - 11?
(9/10, -301/20). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 9/5, product = c/a = -11/5.