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Solve -x² + 3x = 0

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

x² +x += 0
−x2+3x=0⇒x=0, 3-x^2 + 3x = 0 \Rightarrow x = 0,\ 3
Discriminant D = 9 · 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 = -x² + 3x-2-112345-14-12-10-8-6-4-224x = 0x = 3vertexy = -x² + 3x
Graph of y = -x² + 3x. The curve crosses the x-axis at x = 0 and x = 3; the vertex is at (1.5, 2.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = 3, c = 0.

    −x2+3x=0-x^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⋅(−1)⋅0=9+0=9D = 3^2 - 4\cdot(-1)\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:

    −(x)(x−3)=0-\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=−3±9−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{9}}{-2}
  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 = -1·(0) + 3·(0) + 0 = 0 ✓

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

The parabola y = -x² + 3x

xy
-1.5-6.75
-0.5-1.75
0.51.25
1.52.25
2.51.25
3.5-1.75
4.5-6.75

Questions

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

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

Does -x² + 3x = 0 factor?

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

What is the vertex of y = -x² + 3x?

(3/2, 9/4). 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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