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Solve 5x² - 12x = 0

The solutions of 5x² - 12x = 0 are x = 12/5 or x = 0.

x² +x += 0
5x2−12x=0⇒x=125, 05x^2 - 12x = 0 \Rightarrow x = \frac{12}{5},\ 0
Discriminant D = 144 · x ≈ 2.4, 0

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

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Graph of y = 5x² - 12x-2-11234-10102030405060x = 2.4x = 0vertexy = 5x² - 12x
Graph of y = 5x² - 12x. The curve crosses the x-axis at x = 2.4 and x = 0; the vertex is at (1.2, -7.2).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = -12, c = 0.

    5x2−12x=05x^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⋅5⋅0=144+0=144D = (-12)^2 - 4\cdot5\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:

    (5x−12)(x)=0\left(5x - 12\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:

    5x−12=0⇒x=125x=0⇒x=05x - 12 = 0 \Rightarrow x = \frac{12}{5}\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=12±14410x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{144}}{10}
  7. 7

    Simplify to get the solutions:

    x=125orx=0x = \frac{12}{5}\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = 12/5 back in: a·x² + b·x + c = 5·(144/25) + (-12)·(12/5) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 12x = 0\ \Rightarrow\ x = \frac{12}{5},\ 0 \]
\[ 5x^2 - 12x = 0 \]
\[ D = (-12)^2 - 4\cdot5\cdot0 = 144 + 0 = 144 \]
\[ \left(5x - 12\right)\left(x\right) = 0 \]
\[ 5x - 12 = 0 \Rightarrow x = \frac{12}{5}\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{12 \pm \sqrt{144}}{10} \]
\[ x = \frac{12}{5}\quad\text{or}\quad x = 0 \]

The parabola y = 5x² - 12x

xy
-1.837.8
-0.812.8
0.2-2.2
1.2-7.2
2.2-2.2
3.212.8
4.237.8

Questions

What are the solutions of 5x² - 12x = 0?

The solutions of 5x² - 12x = 0 are x = 12/5 or x = 0.

Does 5x² - 12x = 0 factor?

Yes: (5x-12)(x) = 0.

What is the vertex of y = 5x² - 12x?

(6/5, -36/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 12/5, product = c/a = 0.

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