Solve -3x² + 11x - 1 = 0
The solutions of -3x² + 11x - 1 = 0 are x = (11 + √109)/6 or x = (11 - √109)/6, that is x ≈ 0.093282249 and x ≈ 3.5733844.
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 = -3, b = 11, 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 ≈ 0.093282249 and x ≈ 3.5733844.
- 7
Check numerically: for x ≈ 0.093282249 the left side is -6.6613381e-16 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -3x^2 + 11x - 1 = 0\ \Rightarrow\ x = \frac{11 + \sqrt{109}}{6},\ \frac{11 - \sqrt{109}}{6} \]
\[ -3x^2 + 11x - 1 = 0 \]
\[ D = 11^2 - 4\cdot(-3)\cdot(-1) = 121 - 12 = 109 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{109}}{-6} \]
\[ x = \frac{11 + \sqrt{109}}{6}\quad\text{or}\quad x = \frac{11 - \sqrt{109}}{6} \]The parabola y = -3x² + 11x - 1
| x | y |
|---|---|
| -1.1667 | -17.917267 |
| -0.1667 | -2.9170667 |
| 0.8333 | 6.0831333 |
| 1.8333 | 9.0833333 |
| 2.8333 | 6.0835333 |
| 3.8333 | -2.9162667 |
| 4.8333 | -17.916067 |
Questions
What are the solutions of -3x² + 11x - 1 = 0?
The solutions of -3x² + 11x - 1 = 0 are x = (11 + √109)/6 or x = (11 - √109)/6, that is x ≈ 0.093282249 and x ≈ 3.5733844.
Does -3x² + 11x - 1 = 0 factor?
Not over the integers, because the discriminant 109 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -3x² + 11x - 1?
(11/6, 109/12). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 11/3, product = c/a = 1/3.