Solve -3x² + 10x + 9 = 0
The solutions of -3x² + 10x + 9 = 0 are x = (5 + 2√13)/3 or x = (5 - 2√13)/3, that is x ≈ -0.73703418 and x ≈ 4.0703675.
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 = 10, 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 ≈ -0.73703418 and x ≈ 4.0703675.
- 7
Check numerically: for x ≈ -0.73703418 the left side is 1.7763568e-15 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ -3x^2 + 10x + 9 = 0\ \Rightarrow\ x = \frac{5 + 2\sqrt{13}}{3},\ \frac{5 - 2\sqrt{13}}{3} \]
\[ -3x^2 + 10x + 9 = 0 \]
\[ D = 10^2 - 4\cdot(-3)\cdot9 = 100 + 108 = 208 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-10 \pm \sqrt{208}}{-6} \]
\[ x = \frac{5 + 2\sqrt{13}}{3}\quad\text{or}\quad x = \frac{5 - 2\sqrt{13}}{3} \]The parabola y = -3x² + 10x + 9
| x | y |
|---|---|
| -1.3333 | -9.6660667 |
| -0.3333 | 5.3337333 |
| 0.6667 | 14.333533 |
| 1.6667 | 17.333333 |
| 2.6667 | 14.333133 |
| 3.6667 | 5.3329333 |
| 4.6667 | -9.6672667 |
Questions
What are the solutions of -3x² + 10x + 9 = 0?
The solutions of -3x² + 10x + 9 = 0 are x = (5 + 2√13)/3 or x = (5 - 2√13)/3, that is x ≈ -0.73703418 and x ≈ 4.0703675.
Does -3x² + 10x + 9 = 0 factor?
Not over the integers, because the discriminant 208 is not a perfect square. Use the quadratic formula.
What is the vertex of y = -3x² + 10x + 9?
(5/3, 52/3). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 10/3, product = c/a = -3.