Solve -2x² + 11x - 5 = 0
The solutions of -2x² + 11x - 5 = 0 are x = 1/2 or x = 5.
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 = -5.
- 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
Method 1, factoring. The roots are rational, so the left side factors:
- 5
A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:
- 6
Method 2, The quadratic formula works for every quadratic:
- 7
Simplify to get the solutions:
- 8
Check by putting x = 1/2 back in: a·x² + b·x + c = -2·(1/4) + 11·(1/2) + (-5) = 0 ✓
LaTeX source of the formulas on this page
\[ -2x^2 + 11x - 5 = 0\ \Rightarrow\ x = \frac{1}{2},\ 5 \]
\[ -2x^2 + 11x - 5 = 0 \]
\[ D = 11^2 - 4\cdot(-2)\cdot(-5) = 121 - 40 = 81 \]
\[ -\left(2x - 1\right)\left(x - 5\right) = 0 \]
\[ 2x - 1 = 0 \Rightarrow x = \frac{1}{2}\qquad x - 5 = 0 \Rightarrow x = 5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{81}}{-4} \]
\[ x = \frac{1}{2}\quad\text{or}\quad x = 5 \]The parabola y = -2x² + 11x - 5
| x | y |
|---|---|
| -0.25 | -7.875 |
| 0.75 | 2.125 |
| 1.75 | 8.125 |
| 2.75 | 10.125 |
| 3.75 | 8.125 |
| 4.75 | 2.125 |
| 5.75 | -7.875 |
Questions
What are the solutions of -2x² + 11x - 5 = 0?
The solutions of -2x² + 11x - 5 = 0 are x = 1/2 or x = 5.
Does -2x² + 11x - 5 = 0 factor?
Yes: -1(2x-1)(x-5) = 0.
What is the vertex of y = -2x² + 11x - 5?
(11/4, 81/8). 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 = 5/2.