Solve -x² + 12x + 10 = 0
The solutions of -x² + 12x + 10 = 0 are x = 6 + √46 or x = 6 - √46, that is x ≈ -0.78232998 and x ≈ 12.78233.
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 = 12, c = 10.
- 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 ≈ -0.78232998 and x ≈ 12.78233.
- 7
Check numerically: for x ≈ -0.78232998 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -x^2 + 12x + 10 = 0\ \Rightarrow\ x = 6 + \sqrt{46},\ 6 - \sqrt{46} \]
\[ -x^2 + 12x + 10 = 0 \]
\[ D = 12^2 - 4\cdot(-1)\cdot10 = 144 + 40 = 184 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{184}}{-2} \]
\[ x = 6 + \sqrt{46}\quad\text{or}\quad x = 6 - \sqrt{46} \]The parabola y = -x² + 12x + 10
| x | y |
|---|---|
| 3 | 37 |
| 4 | 42 |
| 5 | 45 |
| 6 | 46 |
| 7 | 45 |
| 8 | 42 |
| 9 | 37 |
Questions
What are the solutions of -x² + 12x + 10 = 0?
The solutions of -x² + 12x + 10 = 0 are x = 6 + √46 or x = 6 - √46, that is x ≈ -0.78232998 and x ≈ 12.78233.
Does -x² + 12x + 10 = 0 factor?
Not over the integers, because the discriminant 184 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -x² + 12x + 10?
(6, 46). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 12, product = c/a = -10.