Solve 4x² - 3x - 6 = 0
The solutions of 4x² - 3x - 6 = 0 are x = (3 + √105)/8 or x = (3 - √105)/8, that is x ≈ 1.6558688 and x ≈ -0.90586885.
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 = 4, b = -3, c = -6.
- 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 ≈ 1.6558688 and x ≈ -0.90586885.
- 7
Check numerically: for x ≈ 1.6558688 the left side is -8.8817842e-16 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 4x^2 - 3x - 6 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{105}}{8},\ \frac{3 - \sqrt{105}}{8} \]
\[ 4x^2 - 3x - 6 = 0 \]
\[ D = (-3)^2 - 4\cdot4\cdot(-6) = 9 + 96 = 105 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{105}}{8} \]
\[ x = \frac{3 + \sqrt{105}}{8}\quad\text{or}\quad x = \frac{3 - \sqrt{105}}{8} \]The parabola y = 4x² - 3x - 6
| x | y |
|---|---|
| -2.625 | 29.4375 |
| -1.625 | 9.4375 |
| -0.625 | -2.5625 |
| 0.375 | -6.5625 |
| 1.375 | -2.5625 |
| 2.375 | 9.4375 |
| 3.375 | 29.4375 |
Questions
What are the solutions of 4x² - 3x - 6 = 0?
The solutions of 4x² - 3x - 6 = 0 are x = (3 + √105)/8 or x = (3 - √105)/8, that is x ≈ 1.6558688 and x ≈ -0.90586885.
Does 4x² - 3x - 6 = 0 factor?
Not over the integers, because the discriminant 105 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 4x² - 3x - 6?
(3/8, -105/16). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 3/4, product = c/a = -3/2.