Solve 5x² + 10x - 1 = 0
The solutions of 5x² + 10x - 1 = 0 are x = (-5 + √30)/5 or x = (-5 - √30)/5, that is x ≈ 0.095445115 and x ≈ -2.0954451.
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 = 5, b = 10, 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 ≈ 0.095445115 and x ≈ -2.0954451.
- 7
Check numerically: for x ≈ 0.095445115 the left side is 2.220446e-16 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 5x^2 + 10x - 1 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{30}}{5},\ \frac{-5 - \sqrt{30}}{5} \]
\[ 5x^2 + 10x - 1 = 0 \]
\[ D = 10^2 - 4\cdot5\cdot(-1) = 100 + 20 = 120 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{120}}{10} \]
\[ x = \frac{-5 + \sqrt{30}}{5}\quad\text{or}\quad x = \frac{-5 - \sqrt{30}}{5} \]The parabola y = 5x² + 10x - 1
| x | y |
|---|---|
| -4 | 39 |
| -3 | 14 |
| -2 | -1 |
| -1 | -6 |
| 0 | -1 |
| 1 | 14 |
| 2 | 39 |
Questions
What are the solutions of 5x² + 10x - 1 = 0?
The solutions of 5x² + 10x - 1 = 0 are x = (-5 + √30)/5 or x = (-5 - √30)/5, that is x ≈ 0.095445115 and x ≈ -2.0954451.
Does 5x² + 10x - 1 = 0 factor?
Not over the integers, because the discriminant 120 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 5x² + 10x - 1?
(-1, -6). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -2, product = c/a = -1/5.