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Solve -2x² + 7x = 0

The solutions of -2x² + 7x = 0 are x = 0 or x = 7/2.

x² +x += 0
−2x2+7x=0⇒x=0, 72-2x^2 + 7x = 0 \Rightarrow x = 0,\ \frac{7}{2}
Discriminant D = 49 · x ≈ 0, 3.5

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

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Graph of y = -2x² + 7x-2-1123456-30-25-20-15-10-5510x = 0x = 3.5vertexy = -2x² + 7x
Graph of y = -2x² + 7x. The curve crosses the x-axis at x = 0 and x = 3.5; the vertex is at (1.75, 6.125).

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 = 7, c = 0.

    −2x2+7x=0-2x^2 + 7x = 0
  2. 2

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

    D=72−4⋅(−2)⋅0=49+0=49D = 7^2 - 4\cdot(-2)\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:

    −(x)(2x−7)=0-\left(x\right)\left(2x - 7\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=0⇒x=02x−7=0⇒x=72x = 0 \Rightarrow x = 0\qquad 2x - 7 = 0 \Rightarrow x = \frac{7}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−7±49−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-7 \pm \sqrt{49}}{-4}
  7. 7

    Simplify to get the solutions:

    x=0orx=72x = 0\quad\text{or}\quad x = \frac{7}{2}
  8. 8

    Check by putting x = 0 back in: a·x² + b·x + c = -2·(0) + 7·(0) + 0 = 0 ✓

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

The parabola y = -2x² + 7x

xy
-1.25-11.875
-0.25-1.875
0.754.125
1.756.125
2.754.125
3.75-1.875
4.75-11.875

Questions

What are the solutions of -2x² + 7x = 0?

The solutions of -2x² + 7x = 0 are x = 0 or x = 7/2.

Does -2x² + 7x = 0 factor?

Yes: -1(x)(2x-7) = 0.

What is the vertex of y = -2x² + 7x?

(7/4, 49/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 7/2, product = c/a = 0.

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