Solve 5x² + 6x - 2 = 0
The solutions of 5x² + 6x - 2 = 0 are x = (-3 + √19)/5 or x = (-3 - √19)/5, that is x ≈ 0.27177979 and x ≈ -1.4717798.
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 = 6, c = -2.
- 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 ≈ 0.27177979 and x ≈ -1.4717798.
- 7
Check numerically: for x ≈ 0.27177979 the left side is 8.8817842e-16 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 5x^2 + 6x - 2 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{19}}{5},\ \frac{-3 - \sqrt{19}}{5} \]
\[ 5x^2 + 6x - 2 = 0 \]
\[ D = 6^2 - 4\cdot5\cdot(-2) = 36 + 40 = 76 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{76}}{10} \]
\[ x = \frac{-3 + \sqrt{19}}{5}\quad\text{or}\quad x = \frac{-3 - \sqrt{19}}{5} \]The parabola y = 5x² + 6x - 2
| x | y |
|---|---|
| -3.6 | 41.2 |
| -2.6 | 16.2 |
| -1.6 | 1.2 |
| -0.6 | -3.8 |
| 0.4 | 1.2 |
| 1.4 | 16.2 |
| 2.4 | 41.2 |
Questions
What are the solutions of 5x² + 6x - 2 = 0?
The solutions of 5x² + 6x - 2 = 0 are x = (-3 + √19)/5 or x = (-3 - √19)/5, that is x ≈ 0.27177979 and x ≈ -1.4717798.
Does 5x² + 6x - 2 = 0 factor?
Not over the integers, because the discriminant 76 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 5x² + 6x - 2?
(-3/5, -19/5). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -6/5, product = c/a = -2/5.