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Solve -2x² - 10x - 9 = 0

The solutions of -2x² - 10x - 9 = 0 are x = (-5 + √7)/2 or x = (-5 - √7)/2, that is x ≈ -3.8228757 and x ≈ -1.1771243.

x² +x += 0
−2x2−10x−9=0⇒x=−5+72, −5−72-2x^2 - 10x - 9 = 0 \Rightarrow x = \frac{-5 + \sqrt{7}}{2},\ \frac{-5 - \sqrt{7}}{2}
Discriminant D = 28 · x ≈ -3.82288, -1.17712

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

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Graph of y = -2x² - 10x - 9-6-5-4-3-2-11-25-20-15-10-55x = -3.8228757x = -1.1771243vertexy = -2x² - 10x - 9
Graph of y = -2x² - 10x - 9. The curve crosses the x-axis at x = -3.8228757 and x = -1.1771243; the vertex is at (-2.5, 3.5).

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 = -10, c = -9.

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

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

    D=(−10)2−4⋅(−2)⋅(−9)=100−72=28D = (-10)^2 - 4\cdot(-2)\cdot(-9) = 100 - 72 = 28
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=10±28−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{28}}{-4}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=−5+72orx=−5−72x = \frac{-5 + \sqrt{7}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{7}}{2}
  6. 6

    As decimals: x ≈ -3.8228757 and x ≈ -1.1771243.

  7. 7

    Check numerically: for x ≈ -3.8228757 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 10x - 9 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{7}}{2},\ \frac{-5 - \sqrt{7}}{2} \]
\[ -2x^2 - 10x - 9 = 0 \]
\[ D = (-10)^2 - 4\cdot(-2)\cdot(-9) = 100 - 72 = 28 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{28}}{-4} \]
\[ x = \frac{-5 + \sqrt{7}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{7}}{2} \]

The parabola y = -2x² - 10x - 9

xy
-5.5-14.5
-4.5-4.5
-3.51.5
-2.53.5
-1.51.5
-0.5-4.5
0.5-14.5

Questions

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

The solutions of -2x² - 10x - 9 = 0 are x = (-5 + √7)/2 or x = (-5 - √7)/2, that is x ≈ -3.8228757 and x ≈ -1.1771243.

Does -2x² - 10x - 9 = 0 factor?

Not over the integers, because the discriminant 28 is not a perfect square. Use the quadratic formula.

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

(-5/2, 7/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = 9/2.

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