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Solve x² - 9x = 0

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

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

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

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Graph of y = x² - 9x-2246810-20-10102030x = 9x = 0vertexy = x² - 9x
Graph of y = x² - 9x. The curve crosses the x-axis at x = 9 and x = 0; the vertex is at (4.5, -20.25).

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 = -9, c = 0.

    x2−9x=0x^2 - 9x = 0
  2. 2

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

    D=(−9)2−4⋅1⋅0=81+0=81D = (-9)^2 - 4\cdot1\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:

    (x−9)(x)=0\left(x - 9\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−9=0⇒x=9x=0⇒x=0x - 9 = 0 \Rightarrow x = 9\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=9±812x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{81}}{2}
  7. 7

    Simplify to get the solutions:

    x=9orx=0x = 9\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = 9 back in: a·x² + b·x + c = 1·(81) + (-9)·(9) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - 9x = 0\ \Rightarrow\ x = 9,\ 0 \]
\[ x^2 - 9x = 0 \]
\[ D = (-9)^2 - 4\cdot1\cdot0 = 81 + 0 = 81 \]
\[ \left(x - 9\right)\left(x\right) = 0 \]
\[ x - 9 = 0 \Rightarrow x = 9\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{81}}{2} \]
\[ x = 9\quad\text{or}\quad x = 0 \]

The parabola y = x² - 9x

xy
1.5-11.25
2.5-16.25
3.5-19.25
4.5-20.25
5.5-19.25
6.5-16.25
7.5-11.25

Questions

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

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

Does x² - 9x = 0 factor?

Yes: (x-9)(x) = 0.

What is the vertex of y = x² - 9x?

(9/2, -81/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 9, product = c/a = 0.

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