Solve 3x² - 6x - 12 = 0
The solutions of 3x² - 6x - 12 = 0 are x = 1 + √5 or x = 1 - √5, that is x ≈ 3.236068 and x ≈ -1.236068.
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 = -6, c = -12.
- 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 ≈ 3.236068 and x ≈ -1.236068.
- 7
Check numerically: for x ≈ 3.236068 the left side is 0 (zero up to rounding) ✓
LaTeX source of the formulas on this page
\[ 3x^2 - 6x - 12 = 0\ \Rightarrow\ x = 1 + \sqrt{5},\ 1 - \sqrt{5} \]
\[ 3x^2 - 6x - 12 = 0 \]
\[ D = (-6)^2 - 4\cdot3\cdot(-12) = 36 + 144 = 180 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{180}}{6} \]
\[ x = 1 + \sqrt{5}\quad\text{or}\quad x = 1 - \sqrt{5} \]The parabola y = 3x² - 6x - 12
| x | y |
|---|---|
| -2 | 12 |
| -1 | -3 |
| 0 | -12 |
| 1 | -15 |
| 2 | -12 |
| 3 | -3 |
| 4 | 12 |
Questions
What are the solutions of 3x² - 6x - 12 = 0?
The solutions of 3x² - 6x - 12 = 0 are x = 1 + √5 or x = 1 - √5, that is x ≈ 3.236068 and x ≈ -1.236068.
Does 3x² - 6x - 12 = 0 factor?
Not over the integers, because the discriminant 180 is not a perfect square. Use the quadratic formula.
What is the vertex of y = 3x² - 6x - 12?
(1, -15). The parabola opens upwards, so this is the minimum.
What are the sum and product of the roots?
Sum = −b/a = 2, product = c/a = -4.