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Solve 3x² - 3x = 0

The solutions of 3x² - 3x = 0 are x = 1 or x = 0.

x² +x += 0
3x2−3x=0⇒x=1, 03x^2 - 3x = 0 \Rightarrow x = 1,\ 0
Discriminant D = 9 · x ≈ 1, 0

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

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Graph of y = 3x² - 3x-2-1123510152025x = 1x = 0vertexy = 3x² - 3x
Graph of y = 3x² - 3x. The curve crosses the x-axis at x = 1 and x = 0; the vertex is at (0.5, -0.75).

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 = -3, c = 0.

    3x2−3x=03x^2 - 3x = 0
  2. 2

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

    D=(−3)2−4⋅3⋅0=9+0=9D = (-3)^2 - 4\cdot3\cdot0 = 9 + 0 = 9
  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:

    3(x−1)(x)=03\left(x - 1\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:

    x−1=0⇒x=1x=0⇒x=0x - 1 = 0 \Rightarrow x = 1\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=3±96x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{9}}{6}
  7. 7

    Simplify to get the solutions:

    x=1orx=0x = 1\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = 1 back in: a·x² + b·x + c = 3·(1) + (-3)·(1) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 3x = 0\ \Rightarrow\ x = 1,\ 0 \]
\[ 3x^2 - 3x = 0 \]
\[ D = (-3)^2 - 4\cdot3\cdot0 = 9 + 0 = 9 \]
\[ 3\left(x - 1\right)\left(x\right) = 0 \]
\[ x - 1 = 0 \Rightarrow x = 1\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{9}}{6} \]
\[ x = 1\quad\text{or}\quad x = 0 \]

The parabola y = 3x² - 3x

xy
-2.526.25
-1.511.25
-0.52.25
0.5-0.75
1.52.25
2.511.25
3.526.25

Questions

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

The solutions of 3x² - 3x = 0 are x = 1 or x = 0.

Does 3x² - 3x = 0 factor?

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

What is the vertex of y = 3x² - 3x?

(1/2, -3/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = 0.

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