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Solve x² - 6x = 0

The solutions of x² - 6x = 0 are x = 6 or x = 0.

x² +x += 0
x2−6x=0⇒x=6, 0x^2 - 6x = 0 \Rightarrow x = 6,\ 0
Discriminant D = 36 · x ≈ 6, 0

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

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Graph of y = x² - 6x-22468-10-55101520x = 6x = 0vertexy = x² - 6x
Graph of y = x² - 6x. The curve crosses the x-axis at x = 6 and x = 0; the vertex is at (3, -9).

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 = -6, c = 0.

    x2−6x=0x^2 - 6x = 0
  2. 2

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

    D=(−6)2−4⋅1⋅0=36+0=36D = (-6)^2 - 4\cdot1\cdot0 = 36 + 0 = 36
  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−6)(x)=0\left(x - 6\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−6=0⇒x=6x=0⇒x=0x - 6 = 0 \Rightarrow x = 6\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=6±362x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{36}}{2}
  7. 7

    Simplify to get the solutions:

    x=6orx=0x = 6\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = 6 back in: a·x² + b·x + c = 1·(36) + (-6)·(6) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ x^2 - 6x = 0\ \Rightarrow\ x = 6,\ 0 \]
\[ x^2 - 6x = 0 \]
\[ D = (-6)^2 - 4\cdot1\cdot0 = 36 + 0 = 36 \]
\[ \left(x - 6\right)\left(x\right) = 0 \]
\[ x - 6 = 0 \Rightarrow x = 6\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{36}}{2} \]
\[ x = 6\quad\text{or}\quad x = 0 \]

The parabola y = x² - 6x

xy
00
1-5
2-8
3-9
4-8
5-5
60

Questions

What are the solutions of x² - 6x = 0?

The solutions of x² - 6x = 0 are x = 6 or x = 0.

Does x² - 6x = 0 factor?

Yes: (x-6)(x) = 0.

What is the vertex of y = x² - 6x?

(3, -9). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = 0.

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