Solve 3x² + 8x - 9 = 0
The solutions of 3x² + 8x - 9 = 0 are x = (-4 + √43)/3 or x = (-4 - √43)/3, that is x ≈ 0.85247951 and x ≈ -3.5191462.
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 = 8, 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 ≈ 0.85247951 and x ≈ -3.5191462.
- 7
Check numerically: for x ≈ 0.85247951 the left side is -1.7763568e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 3x^2 + 8x - 9 = 0\ \Rightarrow\ x = \frac{-4 + \sqrt{43}}{3},\ \frac{-4 - \sqrt{43}}{3} \]
\[ 3x^2 + 8x - 9 = 0 \]
\[ D = 8^2 - 4\cdot3\cdot(-9) = 64 + 108 = 172 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{172}}{6} \]
\[ x = \frac{-4 + \sqrt{43}}{3}\quad\text{or}\quad x = \frac{-4 - \sqrt{43}}{3} \]The parabola y = 3x² + 8x - 9
| x | y |
|---|---|
| -4.3333 | 12.666067 |
| -3.3333 | -2.3337333 |
| -2.3333 | -11.333533 |
| -1.3333 | -14.333333 |
| -0.3333 | -11.333133 |
| 0.6667 | -2.3329333 |
| 1.6667 | 12.667267 |
Questions
What are the solutions of 3x² + 8x - 9 = 0?
The solutions of 3x² + 8x - 9 = 0 are x = (-4 + √43)/3 or x = (-4 - √43)/3, that is x ≈ 0.85247951 and x ≈ -3.5191462.
Does 3x² + 8x - 9 = 0 factor?
Not over the integers, because the discriminant 172 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 3x² + 8x - 9?
(-4/3, -43/3). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = -8/3, product = c/a = -3.