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Solve 2x² - 12x = 0

The solutions of 2x² - 12x = 0 are x = 6 or x = 0.

x² +x += 0
2x2−12x=0⇒x=6, 02x^2 - 12x = 0 \Rightarrow x = 6,\ 0
Discriminant D = 144 · 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 = 2x² - 12x-22468-20-1010203040x = 6x = 0vertexy = 2x² - 12x
Graph of y = 2x² - 12x. The curve crosses the x-axis at x = 6 and x = 0; the vertex is at (3, -18).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = -12, c = 0.

    2x2−12x=02x^2 - 12x = 0
  2. 2

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

    D=(−12)2−4⋅2⋅0=144+0=144D = (-12)^2 - 4\cdot2\cdot0 = 144 + 0 = 144
  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:

    2(x−6)(x)=02\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=12±1444x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{144}}{4}
  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 = 2·(36) + (-12)·(6) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 12x = 0\ \Rightarrow\ x = 6,\ 0 \]
\[ 2x^2 - 12x = 0 \]
\[ D = (-12)^2 - 4\cdot2\cdot0 = 144 + 0 = 144 \]
\[ 2\left(x - 6\right)\left(x\right) = 0 \]
\[ x - 6 = 0 \Rightarrow x = 6\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{144}}{4} \]
\[ x = 6\quad\text{or}\quad x = 0 \]

The parabola y = 2x² - 12x

xy
00
1-10
2-16
3-18
4-16
5-10
60

Questions

What are the solutions of 2x² - 12x = 0?

The solutions of 2x² - 12x = 0 are x = 6 or x = 0.

Does 2x² - 12x = 0 factor?

Yes: 2(x-6)(x) = 0.

What is the vertex of y = 2x² - 12x?

(3, -18). 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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