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Solve 3x² - 11x = 0

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

x² +x += 0
3x2−11x=0⇒x=113, 03x^2 - 11x = 0 \Rightarrow x = \frac{11}{3},\ 0
Discriminant D = 121 · x ≈ 3.66667, 0

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

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Graph of y = 3x² - 11x-2-1123456-101020304050x = 3.6666667x = 0vertexy = 3x² - 11x
Graph of y = 3x² - 11x. The curve crosses the x-axis at x = 3.6666667 and x = 0; 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=03x^2 - 11x = 0
  2. 2

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

    D=(−11)2−4⋅3⋅0=121+0=121D = (-11)^2 - 4\cdot3\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:

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

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

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

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±1216x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{121}}{6}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 11/3 back in: a·x² + b·x + c = 3·(121/9) + (-11)·(11/3) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 11x = 0\ \Rightarrow\ x = \frac{11}{3},\ 0 \]
\[ 3x^2 - 11x = 0 \]
\[ D = (-11)^2 - 4\cdot3\cdot0 = 121 + 0 = 121 \]
\[ \left(3x - 11\right)\left(x\right) = 0 \]
\[ 3x - 11 = 0 \Rightarrow x = \frac{11}{3}\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{121}}{6} \]
\[ x = \frac{11}{3}\quad\text{or}\quad x = 0 \]

The parabola y = 3x² - 11x

xy
-1.166716.917267
-0.16671.9170667
0.8333-7.0831333
1.8333-10.083333
2.8333-7.0835333
3.83331.9162667
4.833316.916067

Questions

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

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

Does 3x² - 11x = 0 factor?

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

What is the vertex of y = 3x² - 11x?

(11/6, -121/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 11/3, product = c/a = 0.

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