Solve -3x² - 10x + 8 = 0
The solutions of -3x² - 10x + 8 = 0 are x = -4 or x = 2/3.
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 = 8.
- 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
Method 1, factoring. The roots are rational, so the left side factors:
- 5
A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:
- 6
Method 2, The quadratic formula works for every quadratic:
- 7
Simplify to get the solutions:
- 8
Check by putting x = -4 back in: a·x² + b·x + c = -3·(16) + (-10)·(-4) + 8 = 0 ✓
LaTeX source of the formulas on this page
\[ -3x^2 - 10x + 8 = 0\ \Rightarrow\ x = -4,\ \frac{2}{3} \]
\[ -3x^2 - 10x + 8 = 0 \]
\[ D = (-10)^2 - 4\cdot(-3)\cdot8 = 100 + 96 = 196 \]
\[ -\left(x + 4\right)\left(3x - 2\right) = 0 \]
\[ x + 4 = 0 \Rightarrow x = -4\qquad 3x - 2 = 0 \Rightarrow x = \frac{2}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{196}}{-6} \]
\[ x = -4\quad\text{or}\quad x = \frac{2}{3} \]The parabola y = -3x² - 10x + 8
| x | y |
|---|---|
| -4.6667 | -10.667267 |
| -3.6667 | 4.3329333 |
| -2.6667 | 13.333133 |
| -1.6667 | 16.333333 |
| -0.6667 | 13.333533 |
| 0.3333 | 4.3337333 |
| 1.3333 | -10.666067 |
Questions
What are the solutions of -3x² - 10x + 8 = 0?
The solutions of -3x² - 10x + 8 = 0 are x = -4 or x = 2/3.
Does -3x² - 10x + 8 = 0 factor?
Yes: -1(x+4)(3x-2) = 0.
What is the vertex of y = -3x² - 10x + 8?
(-5/3, 49/3). 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 = -8/3.