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Solve -3x² + 11x = 0

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

x² +x += 0
−3x2+11x=0⇒x=0, 113-3x^2 + 11x = 0 \Rightarrow x = 0,\ \frac{11}{3}
Discriminant D = 121 · x ≈ 0, 3.66667

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

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Graph of y = -3x² + 11x-2-1123456-50-40-30-20-1010x = 0x = 3.6666667vertexy = -3x² + 11x
Graph of y = -3x² + 11x. The curve crosses the x-axis at x = 0 and x = 3.6666667; the vertex is at (1.8333333, 10.083333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = 11, c = 0.

    −3x2+11x=0-3x^2 + 11x = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=112−4⋅(−3)⋅0=121+0=121D = 11^2 - 4\cdot(-3)\cdot0 = 121 + 0 = 121
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(x)(3x−11)=0-\left(x\right)\left(3x - 11\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    x=0⇒x=03x−11=0⇒x=113x = 0 \Rightarrow x = 0\qquad 3x - 11 = 0 \Rightarrow x = \frac{11}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±121−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{121}}{-6}
  7. 7

    Simplify to get the solutions:

    x=0orx=113x = 0\quad\text{or}\quad x = \frac{11}{3}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = -3·(0) + 11·(0) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 11x = 0\ \Rightarrow\ x = 0,\ \frac{11}{3} \]
\[ -3x^2 + 11x = 0 \]
\[ D = 11^2 - 4\cdot(-3)\cdot0 = 121 + 0 = 121 \]
\[ -\left(x\right)\left(3x - 11\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad 3x - 11 = 0 \Rightarrow x = \frac{11}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{121}}{-6} \]
\[ x = 0\quad\text{or}\quad x = \frac{11}{3} \]

The parabola y = -3x² + 11x

xy
-1.1667-16.917267
-0.1667-1.9170667
0.83337.0831333
1.833310.083333
2.83337.0835333
3.8333-1.9162667
4.8333-16.916067

Questions

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

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

Does -3x² + 11x = 0 factor?

Yes: -1(x)(3x-11) = 0.

What is the vertex of y = -3x² + 11x?

(11/6, 121/12). 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 = 0.

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