Solve -2x² + 8x - 1 = 0
The solutions of -2x² + 8x - 1 = 0 are x = (4 + √14)/2 or x = (4 - √14)/2, that is x ≈ 0.12917131 and x ≈ 3.8708287.
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 = 8, c = -1.
- 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 ≈ 0.12917131 and x ≈ 3.8708287.
- 7
Check numerically: for x ≈ 0.12917131 the left side is 2.220446e-16 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -2x^2 + 8x - 1 = 0\ \Rightarrow\ x = \frac{4 + \sqrt{14}}{2},\ \frac{4 - \sqrt{14}}{2} \]
\[ -2x^2 + 8x - 1 = 0 \]
\[ D = 8^2 - 4\cdot(-2)\cdot(-1) = 64 - 8 = 56 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{56}}{-4} \]
\[ x = \frac{4 + \sqrt{14}}{2}\quad\text{or}\quad x = \frac{4 - \sqrt{14}}{2} \]The parabola y = -2x² + 8x - 1
| x | y |
|---|---|
| -1 | -11 |
| 0 | -1 |
| 1 | 5 |
| 2 | 7 |
| 3 | 5 |
| 4 | -1 |
| 5 | -11 |
Questions
What are the solutions of -2x² + 8x - 1 = 0?
The solutions of -2x² + 8x - 1 = 0 are x = (4 + √14)/2 or x = (4 - √14)/2, that is x ≈ 0.12917131 and x ≈ 3.8708287.
Does -2x² + 8x - 1 = 0 factor?
Not over the integers, because the discriminant 56 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -2x² + 8x - 1?
(2, 7). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 4, product = c/a = 1/2.