Solve 3x² - 2x - 11 = 0
The solutions of 3x² - 2x - 11 = 0 are x = (1 + √34)/3 or x = (1 - √34)/3, that is x ≈ 2.276984 and x ≈ -1.6103173.
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 = -2, c = -11.
- 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 ≈ 2.276984 and x ≈ -1.6103173.
- 7
Check numerically: for x ≈ 2.276984 the left side is 1.7763568e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 3x^2 - 2x - 11 = 0\ \Rightarrow\ x = \frac{1 + \sqrt{34}}{3},\ \frac{1 - \sqrt{34}}{3} \]
\[ 3x^2 - 2x - 11 = 0 \]
\[ D = (-2)^2 - 4\cdot3\cdot(-11) = 4 + 132 = 136 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{136}}{6} \]
\[ x = \frac{1 + \sqrt{34}}{3}\quad\text{or}\quad x = \frac{1 - \sqrt{34}}{3} \]The parabola y = 3x² - 2x - 11
| x | y |
|---|---|
| -2.6667 | 15.667267 |
| -1.6667 | 0.66706667 |
| -0.6667 | -8.3331333 |
| 0.3333 | -11.333333 |
| 1.3333 | -8.3335333 |
| 2.3333 | 0.66626667 |
| 3.3333 | 15.666067 |
Questions
What are the solutions of 3x² - 2x - 11 = 0?
The solutions of 3x² - 2x - 11 = 0 are x = (1 + √34)/3 or x = (1 - √34)/3, that is x ≈ 2.276984 and x ≈ -1.6103173.
Does 3x² - 2x - 11 = 0 factor?
Not over the integers, because the discriminant 136 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 3x² - 2x - 11?
(1/3, -34/3). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 2/3, product = c/a = -11/3.