Solve 3x² - 2x - 5 = 0
The solutions of 3x² - 2x - 5 = 0 are x = 5/3 or x = -1.
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 = -5.
- 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 = 5/3 back in: a·x² + b·x + c = 3·(25/9) + (-2)·(5/3) + (-5) = 0 ✓
LaTeX source of the formulas on this page
\[ 3x^2 - 2x - 5 = 0\ \Rightarrow\ x = \frac{5}{3},\ -1 \]
\[ 3x^2 - 2x - 5 = 0 \]
\[ D = (-2)^2 - 4\cdot3\cdot(-5) = 4 + 60 = 64 \]
\[ \left(3x - 5\right)\left(x + 1\right) = 0 \]
\[ 3x - 5 = 0 \Rightarrow x = \frac{5}{3}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{64}}{6} \]
\[ x = \frac{5}{3}\quad\text{or}\quad x = -1 \]The parabola y = 3x² - 2x - 5
| x | y |
|---|---|
| -2.6667 | 21.667267 |
| -1.6667 | 6.6670667 |
| -0.6667 | -2.3331333 |
| 0.3333 | -5.3333333 |
| 1.3333 | -2.3335333 |
| 2.3333 | 6.6662667 |
| 3.3333 | 21.666067 |
Questions
What are the solutions of 3x² - 2x - 5 = 0?
The solutions of 3x² - 2x - 5 = 0 are x = 5/3 or x = -1.
Does 3x² - 2x - 5 = 0 factor?
Yes: (3x-5)(x+1) = 0.
What is the vertex of y = 3x² - 2x - 5?
(1/3, -16/3). 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 = -5/3.