Solve 3x² - 2x - 1 = 0
The solutions of 3x² - 2x - 1 = 0 are x = 1 or x = -1/3.
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 = -2, c = -1.
- 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 = 1 back in: a·x² + b·x + c = 3·(1) + (-2)·(1) + (-1) = 0 ✓
LaTeX source of the formulas on this page
\[ 3x^2 - 2x - 1 = 0\ \Rightarrow\ x = 1,\ -\frac{1}{3} \]
\[ 3x^2 - 2x - 1 = 0 \]
\[ D = (-2)^2 - 4\cdot3\cdot(-1) = 4 + 12 = 16 \]
\[ \left(x - 1\right)\left(3x + 1\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad 3x + 1 = 0 \Rightarrow x = -\frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{16}}{6} \]
\[ x = 1\quad\text{or}\quad x = -\frac{1}{3} \]The parabola y = 3x² - 2x - 1
| x | y |
|---|---|
| -2.6667 | 25.667267 |
| -1.6667 | 10.667067 |
| -0.6667 | 1.6668667 |
| 0.3333 | -1.3333333 |
| 1.3333 | 1.6664667 |
| 2.3333 | 10.666267 |
| 3.3333 | 25.666067 |
Questions
What are the solutions of 3x² - 2x - 1 = 0?
The solutions of 3x² - 2x - 1 = 0 are x = 1 or x = -1/3.
Does 3x² - 2x - 1 = 0 factor?
Yes: (x-1)(3x+1) = 0.
What is the vertex of y = 3x² - 2x - 1?
(1/3, -4/3). 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 = -1/3.