Solve -2x² + 4x + 1 = 0
The solutions of -2x² + 4x + 1 = 0 are x = (2 + √6)/2 or x = (2 - √6)/2, that is x ≈ -0.22474487 and x ≈ 2.2247449.
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 = 4, 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 ≈ -0.22474487 and x ≈ 2.2247449.
- 7
Check numerically: for x ≈ -0.22474487 the left side is 5.5511151e-16 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -2x^2 + 4x + 1 = 0\ \Rightarrow\ x = \frac{2 + \sqrt{6}}{2},\ \frac{2 - \sqrt{6}}{2} \]
\[ -2x^2 + 4x + 1 = 0 \]
\[ D = 4^2 - 4\cdot(-2)\cdot1 = 16 + 8 = 24 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-4 \pm \sqrt{24}}{-4} \]
\[ x = \frac{2 + \sqrt{6}}{2}\quad\text{or}\quad x = \frac{2 - \sqrt{6}}{2} \]The parabola y = -2x² + 4x + 1
| x | y |
|---|---|
| -2 | -15 |
| -1 | -5 |
| 0 | 1 |
| 1 | 3 |
| 2 | 1 |
| 3 | -5 |
| 4 | -15 |
Questions
What are the solutions of -2x² + 4x + 1 = 0?
The solutions of -2x² + 4x + 1 = 0 are x = (2 + √6)/2 or x = (2 - √6)/2, that is x ≈ -0.22474487 and x ≈ 2.2247449.
Does -2x² + 4x + 1 = 0 factor?
Not over the integers, because the discriminant 24 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -2x² + 4x + 1?
(1, 3). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 2, product = c/a = -1/2.