Solve -x² + 8x + 5 = 0
The solutions of -x² + 8x + 5 = 0 are x = 4 + √21 or x = 4 - √21, that is x ≈ -0.58257569 and x ≈ 8.5825757.
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 = -1, b = 8, 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
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
- 6
As decimals: x ≈ -0.58257569 and x ≈ 8.5825757.
- 7
Check numerically: for x ≈ -0.58257569 the left side is 1.7763568e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -x^2 + 8x + 5 = 0\ \Rightarrow\ x = 4 + \sqrt{21},\ 4 - \sqrt{21} \]
\[ -x^2 + 8x + 5 = 0 \]
\[ D = 8^2 - 4\cdot(-1)\cdot5 = 64 + 20 = 84 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{84}}{-2} \]
\[ x = 4 + \sqrt{21}\quad\text{or}\quad x = 4 - \sqrt{21} \]The parabola y = -x² + 8x + 5
| x | y |
|---|---|
| 1 | 12 |
| 2 | 17 |
| 3 | 20 |
| 4 | 21 |
| 5 | 20 |
| 6 | 17 |
| 7 | 12 |
Questions
What are the solutions of -x² + 8x + 5 = 0?
The solutions of -x² + 8x + 5 = 0 are x = 4 + √21 or x = 4 - √21, that is x ≈ -0.58257569 and x ≈ 8.5825757.
Does -x² + 8x + 5 = 0 factor?
Not over the integers, because the discriminant 84 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -x² + 8x + 5?
(4, 21). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 8, product = c/a = -5.