Solve 3x² - x - 4 = 0
The solutions of 3x² - x - 4 = 0 are x = 4/3 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 = -1, c = -4.
- 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 = 4/3 back in: a·x² + b·x + c = 3·(16/9) + (-1)·(4/3) + (-4) = 0 ✓
LaTeX source of the formulas on this page
\[ 3x^2 - x - 4 = 0\ \Rightarrow\ x = \frac{4}{3},\ -1 \]
\[ 3x^2 - x - 4 = 0 \]
\[ D = (-1)^2 - 4\cdot3\cdot(-4) = 1 + 48 = 49 \]
\[ \left(3x - 4\right)\left(x + 1\right) = 0 \]
\[ 3x - 4 = 0 \Rightarrow x = \frac{4}{3}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{49}}{6} \]
\[ x = \frac{4}{3}\quad\text{or}\quad x = -1 \]The parabola y = 3x² - x - 4
| x | y |
|---|---|
| -2.8333 | 22.916067 |
| -1.8333 | 7.9162667 |
| -0.8333 | -1.0835333 |
| 0.1667 | -4.0833333 |
| 1.1667 | -1.0831333 |
| 2.1667 | 7.9170667 |
| 3.1667 | 22.917267 |
Questions
What are the solutions of 3x² - x - 4 = 0?
The solutions of 3x² - x - 4 = 0 are x = 4/3 or x = -1.
Does 3x² - x - 4 = 0 factor?
Yes: (3x-4)(x+1) = 0.
What is the vertex of y = 3x² - x - 4?
(1/6, -49/12). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 1/3, product = c/a = -4/3.