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Solve -3x² + 3x = 0

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

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

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

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Graph of y = -3x² + 3x-2-1123-25-20-15-10-5x = 0x = 1vertexy = -3x² + 3x
Graph of y = -3x² + 3x. The curve crosses the x-axis at x = 0 and x = 1; 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=0-3x^2 + 3x = 0
  2. 2

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

    D=32−4⋅(−3)⋅0=9+0=9D = 3^2 - 4\cdot(-3)\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)(x−1)=0-3\left(x\right)\left(x - 1\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=0x−1=0⇒x=1x = 0 \Rightarrow x = 0\qquad x - 1 = 0 \Rightarrow x = 1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

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

    Simplify to get the solutions:

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

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

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

The parabola y = -3x² + 3x

xy
-2.5-26.25
-1.5-11.25
-0.5-2.25
0.50.75
1.5-2.25
2.5-11.25
3.5-26.25

Questions

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

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

Does -3x² + 3x = 0 factor?

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

What is the vertex of y = -3x² + 3x?

(1/2, 3/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = 0.

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