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Solve 4x² - 10x = 0

The solutions of 4x² - 10x = 0 are x = 5/2 or x = 0.

x² +x += 0
4x2−10x=0⇒x=52, 04x^2 - 10x = 0 \Rightarrow x = \frac{5}{2},\ 0
Discriminant D = 100 · x ≈ 2.5, 0

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

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Graph of y = 4x² - 10x-2-112345-101020304050x = 2.5x = 0vertexy = 4x² - 10x
Graph of y = 4x² - 10x. The curve crosses the x-axis at x = 2.5 and x = 0; the vertex is at (1.25, -6.25).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 4, b = -10, c = 0.

    4x2−10x=04x^2 - 10x = 0
  2. 2

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

    D=(−10)2−4⋅4⋅0=100+0=100D = (-10)^2 - 4\cdot4\cdot0 = 100 + 0 = 100
  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(2x−5)(x)=02\left(2x - 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:

    2x−5=0⇒x=52x=0⇒x=02x - 5 = 0 \Rightarrow x = \frac{5}{2}\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±1008x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{100}}{8}
  7. 7

    Simplify to get the solutions:

    x=52orx=0x = \frac{5}{2}\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = 5/2 back in: a·x² + b·x + c = 4·(25/4) + (-10)·(5/2) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 4x^2 - 10x = 0\ \Rightarrow\ x = \frac{5}{2},\ 0 \]
\[ 4x^2 - 10x = 0 \]
\[ D = (-10)^2 - 4\cdot4\cdot0 = 100 + 0 = 100 \]
\[ 2\left(2x - 5\right)\left(x\right) = 0 \]
\[ 2x - 5 = 0 \Rightarrow x = \frac{5}{2}\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{100}}{8} \]
\[ x = \frac{5}{2}\quad\text{or}\quad x = 0 \]

The parabola y = 4x² - 10x

xy
-1.7529.75
-0.759.75
0.25-2.25
1.25-6.25
2.25-2.25
3.259.75
4.2529.75

Questions

What are the solutions of 4x² - 10x = 0?

The solutions of 4x² - 10x = 0 are x = 5/2 or x = 0.

Does 4x² - 10x = 0 factor?

Yes: 2(2x-5)(x) = 0.

What is the vertex of y = 4x² - 10x?

(5/4, -25/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5/2, product = c/a = 0.

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