Solve 3x² - 4x - 1 = 0
The solutions of 3x² - 4x - 1 = 0 are x = (2 + √7)/3 or x = (2 - √7)/3, that is x ≈ 1.5485838 and x ≈ -0.21525044.
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 = -4, 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
The quadratic formula works for every quadratic:
- 5
Simplify (pull the square out of the root) to get the solutions:
- 6
As decimals: x ≈ 1.5485838 and x ≈ -0.21525044.
- 7
Check numerically: for x ≈ 1.5485838 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 3x^2 - 4x - 1 = 0\ \Rightarrow\ x = \frac{2 + \sqrt{7}}{3},\ \frac{2 - \sqrt{7}}{3} \]
\[ 3x^2 - 4x - 1 = 0 \]
\[ D = (-4)^2 - 4\cdot3\cdot(-1) = 16 + 12 = 28 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{4 \pm \sqrt{28}}{6} \]
\[ x = \frac{2 + \sqrt{7}}{3}\quad\text{or}\quad x = \frac{2 - \sqrt{7}}{3} \]The parabola y = 3x² - 4x - 1
| x | y |
|---|---|
| -2.3333 | 24.666067 |
| -1.3333 | 9.6662667 |
| -0.3333 | 0.66646667 |
| 0.6667 | -2.3333333 |
| 1.6667 | 0.66686667 |
| 2.6667 | 9.6670667 |
| 3.6667 | 24.667267 |
Questions
What are the solutions of 3x² - 4x - 1 = 0?
The solutions of 3x² - 4x - 1 = 0 are x = (2 + √7)/3 or x = (2 - √7)/3, that is x ≈ 1.5485838 and x ≈ -0.21525044.
Does 3x² - 4x - 1 = 0 factor?
Not over the integers, because the discriminant 28 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 3x² - 4x - 1?
(2/3, -7/3). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 4/3, product = c/a = -1/3.