Solve 3x² - 4x - 9 = 0
The solutions of 3x² - 4x - 9 = 0 are x = (2 + √31)/3 or x = (2 - √31)/3, that is x ≈ 2.5225881 and x ≈ -1.1892548.
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 = -4, c = -9.
- 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 ≈ 2.5225881 and x ≈ -1.1892548.
- 7
Check numerically: for x ≈ 2.5225881 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 3x^2 - 4x - 9 = 0\ \Rightarrow\ x = \frac{2 + \sqrt{31}}{3},\ \frac{2 - \sqrt{31}}{3} \]
\[ 3x^2 - 4x - 9 = 0 \]
\[ D = (-4)^2 - 4\cdot3\cdot(-9) = 16 + 108 = 124 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{124}}{6} \]
\[ x = \frac{2 + \sqrt{31}}{3}\quad\text{or}\quad x = \frac{2 - \sqrt{31}}{3} \]The parabola y = 3x² - 4x - 9
| x | y |
|---|---|
| -2.3333 | 16.666067 |
| -1.3333 | 1.6662667 |
| -0.3333 | -7.3335333 |
| 0.6667 | -10.333333 |
| 1.6667 | -7.3331333 |
| 2.6667 | 1.6670667 |
| 3.6667 | 16.667267 |
Questions
What are the solutions of 3x² - 4x - 9 = 0?
The solutions of 3x² - 4x - 9 = 0 are x = (2 + √31)/3 or x = (2 - √31)/3, that is x ≈ 2.5225881 and x ≈ -1.1892548.
Does 3x² - 4x - 9 = 0 factor?
Not over the integers, because the discriminant 124 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 3x² - 4x - 9?
(2/3, -31/3). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 4/3, product = c/a = -3.