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Solve -2x² - 11x - 10 = 0

The solutions of -2x² - 11x - 10 = 0 are x = (-11 + √41)/4 or x = (-11 - √41)/4, that is x ≈ -4.3507811 and x ≈ -1.1492189.

x² +x += 0
−2x2−11x−10=0⇒x=−11+414, −11−414-2x^2 - 11x - 10 = 0 \Rightarrow x = \frac{-11 + \sqrt{41}}{4},\ \frac{-11 - \sqrt{41}}{4}
Discriminant D = 41 · x ≈ -4.35078, -1.14922

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

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Graph of y = -2x² - 11x - 10-6-5-4-3-2-11-30-25-20-15-10-55x = -4.3507811x = -1.1492189vertexy = -2x² - 11x - 10
Graph of y = -2x² - 11x - 10. The curve crosses the x-axis at x = -4.3507811 and x = -1.1492189; the vertex is at (-2.75, 5.125).

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 = -11, c = -10.

    −2x2−11x−10=0-2x^2 - 11x - 10 = 0
  2. 2

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

    D=(−11)2−4⋅(−2)⋅(−10)=121−80=41D = (-11)^2 - 4\cdot(-2)\cdot(-10) = 121 - 80 = 41
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=11±41−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{41}}{-4}
  5. 5

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

    x=−11+414orx=−11−414x = \frac{-11 + \sqrt{41}}{4}\quad\text{or}\quad x = \frac{-11 - \sqrt{41}}{4}
  6. 6

    As decimals: x ≈ -4.3507811 and x ≈ -1.1492189.

  7. 7

    Check numerically: for x ≈ -4.3507811 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 11x - 10 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{41}}{4},\ \frac{-11 - \sqrt{41}}{4} \]
\[ -2x^2 - 11x - 10 = 0 \]
\[ D = (-11)^2 - 4\cdot(-2)\cdot(-10) = 121 - 80 = 41 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{41}}{-4} \]
\[ x = \frac{-11 + \sqrt{41}}{4}\quad\text{or}\quad x = \frac{-11 - \sqrt{41}}{4} \]

The parabola y = -2x² - 11x - 10

xy
-5.75-12.875
-4.75-2.875
-3.753.125
-2.755.125
-1.753.125
-0.75-2.875
0.25-12.875

Questions

What are the solutions of -2x² - 11x - 10 = 0?

The solutions of -2x² - 11x - 10 = 0 are x = (-11 + √41)/4 or x = (-11 - √41)/4, that is x ≈ -4.3507811 and x ≈ -1.1492189.

Does -2x² - 11x - 10 = 0 factor?

Not over the integers, because the discriminant 41 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 11x - 10?

(-11/4, 41/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -11/2, product = c/a = 5.

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    Solve -2x² - 11x - 10 = 0: x = (-11 + √41)/4 or x = (-11 - √41)/4 (steps) | WordGPT