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Solve 5x² - 7x = 0

The solutions of 5x² - 7x = 0 are x = 7/5 or x = 0.

x² +x += 0
5x2−7x=0⇒x=75, 05x^2 - 7x = 0 \Rightarrow x = \frac{7}{5},\ 0
Discriminant D = 49 · x ≈ 1.4, 0

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

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Graph of y = 5x² - 7x-2-11231020304050x = 1.4x = 0vertexy = 5x² - 7x
Graph of y = 5x² - 7x. The curve crosses the x-axis at x = 1.4 and x = 0; the vertex is at (0.7, -2.45).

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 = -7, c = 0.

    5x2−7x=05x^2 - 7x = 0
  2. 2

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

    D=(−7)2−4⋅5⋅0=49+0=49D = (-7)^2 - 4\cdot5\cdot0 = 49 + 0 = 49
  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−7)(x)=0\left(5x - 7\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−7=0⇒x=75x=0⇒x=05x - 7 = 0 \Rightarrow x = \frac{7}{5}\qquad x = 0 \Rightarrow x = 0
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=7±4910x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{49}}{10}
  7. 7

    Simplify to get the solutions:

    x=75orx=0x = \frac{7}{5}\quad\text{or}\quad x = 0
  8. 8

    Check by putting x = 7/5 back in: a·x² + b·x + c = 5·(49/25) + (-7)·(7/5) + 0 = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 7x = 0\ \Rightarrow\ x = \frac{7}{5},\ 0 \]
\[ 5x^2 - 7x = 0 \]
\[ D = (-7)^2 - 4\cdot5\cdot0 = 49 + 0 = 49 \]
\[ \left(5x - 7\right)\left(x\right) = 0 \]
\[ 5x - 7 = 0 \Rightarrow x = \frac{7}{5}\qquad x = 0 \Rightarrow x = 0 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{7 \pm \sqrt{49}}{10} \]
\[ x = \frac{7}{5}\quad\text{or}\quad x = 0 \]

The parabola y = 5x² - 7x

xy
-2.342.55
-1.317.55
-0.32.55
0.7-2.45
1.72.55
2.717.55
3.742.55

Questions

What are the solutions of 5x² - 7x = 0?

The solutions of 5x² - 7x = 0 are x = 7/5 or x = 0.

Does 5x² - 7x = 0 factor?

Yes: (5x-7)(x) = 0.

What is the vertex of y = 5x² - 7x?

(7/10, -49/20). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 7/5, product = c/a = 0.

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