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Solve -3x² + 9x = 0

The solutions of -3x² + 9x = 0 are x = 0 or x = 3.

x² +x += 0
−3x2+9x=0⇒x=0, 3-3x^2 + 9x = 0 \Rightarrow x = 0,\ 3
Discriminant D = 81 · x ≈ 0, 3

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

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Graph of y = -3x² + 9x-2-112345-40-30-20-1010x = 0x = 3vertexy = -3x² + 9x
Graph of y = -3x² + 9x. The curve crosses the x-axis at x = 0 and x = 3; the vertex is at (1.5, 6.75).

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 = 9, c = 0.

    −3x2+9x=0-3x^2 + 9x = 0
  2. 2

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

    D=92−4⋅(−3)⋅0=81+0=81D = 9^2 - 4\cdot(-3)\cdot0 = 81 + 0 = 81
  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−3)=0-3\left(x\right)\left(x - 3\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−3=0⇒x=3x = 0 \Rightarrow x = 0\qquad x - 3 = 0 \Rightarrow x = 3
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−9±81−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{81}}{-6}
  7. 7

    Simplify to get the solutions:

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

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

LaTeX source of the formulas on this page
\[ -3x^2 + 9x = 0\ \Rightarrow\ x = 0,\ 3 \]
\[ -3x^2 + 9x = 0 \]
\[ D = 9^2 - 4\cdot(-3)\cdot0 = 81 + 0 = 81 \]
\[ -3\left(x\right)\left(x - 3\right) = 0 \]
\[ x = 0 \Rightarrow x = 0\qquad x - 3 = 0 \Rightarrow x = 3 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{81}}{-6} \]
\[ x = 0\quad\text{or}\quad x = 3 \]

The parabola y = -3x² + 9x

xy
-1.5-20.25
-0.5-5.25
0.53.75
1.56.75
2.53.75
3.5-5.25
4.5-20.25

Questions

What are the solutions of -3x² + 9x = 0?

The solutions of -3x² + 9x = 0 are x = 0 or x = 3.

Does -3x² + 9x = 0 factor?

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

What is the vertex of y = -3x² + 9x?

(3/2, 27/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = 0.

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