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Solve -2x² - 9x - 10 = 0

The solutions of -2x² - 9x - 10 = 0 are x = -5/2 or x = -2.

x² +x += 0
−2x2−9x−10=0⇒x=−52, −2-2x^2 - 9x - 10 = 0 \Rightarrow x = -\frac{5}{2},\ -2
Discriminant D = 1 · 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 - 9x - 10-5-4-3-2-1-16-14-12-10-8-6-4-22x = -2.5x = -2vertexy = -2x² - 9x - 10
Graph of y = -2x² - 9x - 10. The curve crosses the x-axis at x = -2.5 and x = -2; the vertex is at (-2.25, 0.125).

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 = -9, c = -10.

    −2x2−9x−10=0-2x^2 - 9x - 10 = 0
  2. 2

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

    D=(−9)2−4⋅(−2)⋅(−10)=81−80=1D = (-9)^2 - 4\cdot(-2)\cdot(-10) = 81 - 80 = 1
  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=9±1−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{1}}{-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) + (-9)·(-5/2) + (-10) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 9x - 10 = 0\ \Rightarrow\ x = -\frac{5}{2},\ -2 \]
\[ -2x^2 - 9x - 10 = 0 \]
\[ D = (-9)^2 - 4\cdot(-2)\cdot(-10) = 81 - 80 = 1 \]
\[ -\left(2x + 5\right)\left(x + 2\right) = 0 \]
\[ 2x + 5 = 0 \Rightarrow x = -\frac{5}{2}\qquad x + 2 = 0 \Rightarrow x = -2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{9 \pm \sqrt{1}}{-4} \]
\[ x = -\frac{5}{2}\quad\text{or}\quad x = -2 \]

The parabola y = -2x² - 9x - 10

xy
-5.25-17.875
-4.25-7.875
-3.25-1.875
-2.250.125
-1.25-1.875
-0.25-7.875
0.75-17.875

Questions

What are the solutions of -2x² - 9x - 10 = 0?

The solutions of -2x² - 9x - 10 = 0 are x = -5/2 or x = -2.

Does -2x² - 9x - 10 = 0 factor?

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

What is the vertex of y = -2x² - 9x - 10?

(-9/4, 1/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -9/2, product = c/a = 5.

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