Solve -3x² + 12x - 12 = 0
The solution of -3x² + 12x - 12 = 0 is x = 2 (a repeated root).
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 = 12, c = -12.
- 2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
- 3
D is zero, so there is exactly one solution (a repeated root).
- 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 solution:
- 8
Check by putting x = 2 back in: a·x² + b·x + c = -3·(4) + 12·(2) + (-12) = 0 ✓
LaTeX source of the formulas on this page
\[ -3x^2 + 12x - 12 = 0\ \Rightarrow\ x = 2 \]
\[ -3x^2 + 12x - 12 = 0 \]
\[ D = 12^2 - 4\cdot(-3)\cdot(-12) = 144 - 144 = 0 \]
\[ -3\left(x - 2\right)\left(x - 2\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-12 \pm \sqrt{0}}{-6} \]
\[ x = 2 \]The parabola y = -3x² + 12x - 12
| x | y |
|---|---|
| -1 | -27 |
| 0 | -12 |
| 1 | -3 |
| 2 | 0 |
| 3 | -3 |
| 4 | -12 |
| 5 | -27 |
Questions
What are the solutions of -3x² + 12x - 12 = 0?
The solution of -3x² + 12x - 12 = 0 is x = 2 (a repeated root).
Does -3x² + 12x - 12 = 0 factor?
Yes: -3(x-2)(x-2) = 0.
What is the vertex of y = -3x² + 12x - 12?
(2, 0). The parabola opens downwards, so this is the maximum.
What are the sum and product of the roots?
Sum = −b/a = 4, product = c/a = 4.