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Solve -2x² - 10x - 6 = 0

The solutions of -2x² - 10x - 6 = 0 are x = (-5 + √13)/2 or x = (-5 - √13)/2, that is x ≈ -4.3027756 and x ≈ -0.69722436.

x² +x += 0
−2x2−10x−6=0⇒x=−5+132, −5−132-2x^2 - 10x - 6 = 0 \Rightarrow x = \frac{-5 + \sqrt{13}}{2},\ \frac{-5 - \sqrt{13}}{2}
Discriminant D = 52 · x ≈ -4.30278, -0.697224

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

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Graph of y = -2x² - 10x - 6-6-5-4-3-2-11-30-25-20-15-10-5510x = -4.3027756x = -0.69722436vertexy = -2x² - 10x - 6
Graph of y = -2x² - 10x - 6. The curve crosses the x-axis at x = -4.3027756 and x = -0.69722436; the vertex is at (-2.5, 6.5).

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 = -6.

    −2x2−10x−6=0-2x^2 - 10x - 6 = 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)⋅(−6)=100−48=52D = (-10)^2 - 4\cdot(-2)\cdot(-6) = 100 - 48 = 52
  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±52−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{52}}{-4}
  5. 5

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

    x=−5+132orx=−5−132x = \frac{-5 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{13}}{2}
  6. 6

    As decimals: x ≈ -4.3027756 and x ≈ -0.69722436.

  7. 7

    Check numerically: for x ≈ -4.3027756 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 10x - 6 = 0\ \Rightarrow\ x = \frac{-5 + \sqrt{13}}{2},\ \frac{-5 - \sqrt{13}}{2} \]
\[ -2x^2 - 10x - 6 = 0 \]
\[ D = (-10)^2 - 4\cdot(-2)\cdot(-6) = 100 - 48 = 52 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{52}}{-4} \]
\[ x = \frac{-5 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{-5 - \sqrt{13}}{2} \]

The parabola y = -2x² - 10x - 6

xy
-5.5-11.5
-4.5-1.5
-3.54.5
-2.56.5
-1.54.5
-0.5-1.5
0.5-11.5

Questions

What are the solutions of -2x² - 10x - 6 = 0?

The solutions of -2x² - 10x - 6 = 0 are x = (-5 + √13)/2 or x = (-5 - √13)/2, that is x ≈ -4.3027756 and x ≈ -0.69722436.

Does -2x² - 10x - 6 = 0 factor?

Not over the integers, because the discriminant 52 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 10x - 6?

(-5/2, 13/2). 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 = 3.

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