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Solve -2x² - x + 10 = 0

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

x² +x += 0
−2x2−x+10=0⇒x=−52, 2-2x^2 - x + 10 = 0 \Rightarrow x = -\frac{5}{2},\ 2
Discriminant D = 81 · x ≈ -2.5, 2

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

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Graph of y = -2x² - x + 10-5-4-3-2-11234-30-20-1010x = -2.5x = 2vertexy = -2x² - x + 10
Graph of y = -2x² - x + 10. The curve crosses the x-axis at x = -2.5 and x = 2; the vertex is at (-0.25, 10.125).

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 = -1, c = 10.

    −2x2−x+10=0-2x^2 - x + 10 = 0
  2. 2

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

    D=(−1)2−4⋅(−2)⋅10=1+80=81D = (-1)^2 - 4\cdot(-2)\cdot10 = 1 + 80 = 81
  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:

    −(2x+5)(x−2)=0-\left(2x + 5\right)\left(x - 2\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−2=0⇒x=22x + 5 = 0 \Rightarrow x = -\frac{5}{2}\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±81−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{81}}{-4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -5/2 back in: a·x² + b·x + c = -2·(25/4) + (-1)·(-5/2) + 10 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - x + 10 = 0\ \Rightarrow\ x = -\frac{5}{2},\ 2 \]
\[ -2x^2 - x + 10 = 0 \]
\[ D = (-1)^2 - 4\cdot(-2)\cdot10 = 1 + 80 = 81 \]
\[ -\left(2x + 5\right)\left(x - 2\right) = 0 \]
\[ 2x + 5 = 0 \Rightarrow x = -\frac{5}{2}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{81}}{-4} \]
\[ x = -\frac{5}{2}\quad\text{or}\quad x = 2 \]

The parabola y = -2x² - x + 10

xy
-3.25-7.875
-2.252.125
-1.258.125
-0.2510.125
0.758.125
1.752.125
2.75-7.875

Questions

What are the solutions of -2x² - x + 10 = 0?

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

Does -2x² - x + 10 = 0 factor?

Yes: -1(2x+5)(x-2) = 0.

What is the vertex of y = -2x² - x + 10?

(-1/4, 81/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -1/2, product = c/a = -5.

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