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Solve 3x² - 5x = 0

The solutions of 3x² - 5x = 0 are x = 5/3 or x = 0.

x² +x += 0
3x2−5x=0⇒x=53, 03x^2 - 5x = 0 \Rightarrow x = \frac{5}{3},\ 0
Discriminant D = 25 · x ≈ 1.66667, 0

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

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Graph of y = 3x² - 5x-2-11234-551015202530x = 1.6666667x = 0vertexy = 3x² - 5x
Graph of y = 3x² - 5x. The curve crosses the x-axis at x = 1.6666667 and x = 0; the vertex is at (0.83333333, -2.0833333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 3, b = -5, c = 0.

    3x2−5x=03x^2 - 5x = 0
  2. 2

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

    D=(−5)2−4⋅3⋅0=25+0=25D = (-5)^2 - 4\cdot3\cdot0 = 25 + 0 = 25
  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:

    (3x−5)(x)=0\left(3x - 5\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:

    3x−5=0⇒x=53x=0⇒x=03x - 5 = 0 \Rightarrow x = \frac{5}{3}\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=5±256x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{25}}{6}
  7. 7

    Simplify to get the solutions:

    x=53orx=0x = \frac{5}{3}\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = 5/3 back in: a·x² + b·x + c = 3·(25/9) + (-5)·(5/3) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 5x = 0\ \Rightarrow\ x = \frac{5}{3},\ 0 \]
\[ 3x^2 - 5x = 0 \]
\[ D = (-5)^2 - 4\cdot3\cdot0 = 25 + 0 = 25 \]
\[ \left(3x - 5\right)\left(x\right) = 0 \]
\[ 3x - 5 = 0 \Rightarrow x = \frac{5}{3}\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{25}}{6} \]
\[ x = \frac{5}{3}\quad\text{or}\quad x = 0 \]

The parabola y = 3x² - 5x

xy
-2.166724.917267
-1.16679.9170667
-0.16670.91686667
0.8333-2.0833333
1.83330.91646667
2.83339.9162667
3.833324.916067

Questions

What are the solutions of 3x² - 5x = 0?

The solutions of 3x² - 5x = 0 are x = 5/3 or x = 0.

Does 3x² - 5x = 0 factor?

Yes: (3x-5)(x) = 0.

What is the vertex of y = 3x² - 5x?

(5/6, -25/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5/3, product = c/a = 0.

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