Solve -3x² + 11x - 6 = 0

The solutions of -3x² + 11x - 6 = 0 are x = 2/3 or x = 3.

x² +x += 0

-3x^2 + 11x - 6 = 0 \Rightarrow x = \frac{2}{3},\ 3

Discriminant D = 49 · x ≈ 0.666667, 3

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

-3x^2 + 11x - 6 = 0\ \Rightarrow\ x = \frac{2}{3},\ 3

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Graph of y = -3x² + 11x - 6. The curve crosses the x-axis at x = 0.66666667 and x = 3; the vertex is at (1.8333333, 4.0833333).

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 = 11, c = -6.
$-3x^2 + 11x - 6 = 0$
2
Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.
$D = 11^2 - 4\cdot(-3)\cdot(-6) = 121 - 72 = 49$
3
D is positive, so there are two different real solutions.
4
Method 1, factoring. The roots are rational, so the left side factors:
$-\left(3x - 2\right)\left(x - 3\right) = 0$
5
A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:
$3x - 2 = 0 \Rightarrow x = \frac{2}{3}\qquad x - 3 = 0 \Rightarrow x = 3$
6
Method 2, The quadratic formula works for every quadratic:
$x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{49}}{-6}$
7
Simplify to get the solutions:
$x = \frac{2}{3}\quad\text{or}\quad x = 3$
8
Check by putting x = 2/3 back in: a·x² + b·x + c = -3·(4/9) + 11·(2/3) + (-6) = 0 ✓

LaTeX source of the formulas on this page

\[ -3x^2 + 11x - 6 = 0\ \Rightarrow\ x = \frac{2}{3},\ 3 \]
\[ -3x^2 + 11x - 6 = 0 \]
\[ D = 11^2 - 4\cdot(-3)\cdot(-6) = 121 - 72 = 49 \]
\[ -\left(3x - 2\right)\left(x - 3\right) = 0 \]
\[ 3x - 2 = 0 \Rightarrow x = \frac{2}{3}\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{49}}{-6} \]
\[ x = \frac{2}{3}\quad\text{or}\quad x = 3 \]

The parabola y = -3x² + 11x - 6

x	y
-1.1667	-22.917267
-0.1667	-7.9170667
0.8333	1.0831333
1.8333	4.0833333
2.8333	1.0835333
3.8333	-7.9162667
4.8333	-22.916067

Questions

What are the solutions of -3x² + 11x - 6 = 0?

The solutions of -3x² + 11x - 6 = 0 are x = 2/3 or x = 3.

Does -3x² + 11x - 6 = 0 factor?

Yes: -1(3x-2)(x-3) = 0.

What is the vertex of y = -3x² + 11x - 6?

(11/6, 49/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 11/3, product = c/a = 2.

Solve -3x² + 11x - 6 = 0

How to work it out

The parabola y = -3x² + 11x - 6

Questions

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