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Solve -2x² - 10x - 5 = 0

The solutions of -2x² - 10x - 5 = 0 are x = (-5 + √15)/2 or x = (-5 - √15)/2, that is x ≈ -4.4364917 and x ≈ -0.56350833.

x² +x += 0
−2x2−10x−5=0⇒x=−5+152, −5−152-2x^2 - 10x - 5 = 0 \Rightarrow x = \frac{-5 + \sqrt{15}}{2},\ \frac{-5 - \sqrt{15}}{2}
Discriminant D = 60 · x ≈ -4.43649, -0.563508

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

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Graph of y = -2x² - 10x - 5-6-5-4-3-2-11-30-20-1010x = -4.4364917x = -0.56350833vertexy = -2x² - 10x - 5
Graph of y = -2x² - 10x - 5. The curve crosses the x-axis at x = -4.4364917 and x = -0.56350833; the vertex is at (-2.5, 7.5).

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 = -5.

    −2x2−10x−5=0-2x^2 - 10x - 5 = 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)⋅(−5)=100−40=60D = (-10)^2 - 4\cdot(-2)\cdot(-5) = 100 - 40 = 60
  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±60−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{60}}{-4}
  5. 5

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

    x=−5+152orx=−5−152x = \frac{-5 + \sqrt{15}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{15}}{2}
  6. 6

    As decimals: x ≈ -4.4364917 and x ≈ -0.56350833.

  7. 7

    Check numerically: for x ≈ -4.4364917 the left side is 7.1054274e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 10x - 5 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{15}}{2},\ \frac{-5 - \sqrt{15}}{2} \]
\[ -2x^2 - 10x - 5 = 0 \]
\[ D = (-10)^2 - 4\cdot(-2)\cdot(-5) = 100 - 40 = 60 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{60}}{-4} \]
\[ x = \frac{-5 + \sqrt{15}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{15}}{2} \]

The parabola y = -2x² - 10x - 5

xy
-5.5-10.5
-4.5-0.5
-3.55.5
-2.57.5
-1.55.5
-0.5-0.5
0.5-10.5

Questions

What are the solutions of -2x² - 10x - 5 = 0?

The solutions of -2x² - 10x - 5 = 0 are x = (-5 + √15)/2 or x = (-5 - √15)/2, that is x ≈ -4.4364917 and x ≈ -0.56350833.

Does -2x² - 10x - 5 = 0 factor?

Not over the integers, because the discriminant 60 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 10x - 5?

(-5/2, 15/2). 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 = 5/2.

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    Solve -2x² - 10x - 5 = 0: x = (-5 + √15)/2 or x = (-5 - √15)/2 (steps) | WordGPT