WordGPT

Solve -2x² - 11x - 8 = 0

The solutions of -2x² - 11x - 8 = 0 are x = (-11 + √57)/4 or x = (-11 - √57)/4, that is x ≈ -4.6374586 and x ≈ -0.86254139.

x² +x += 0
−2x2−11x−8=0⇒x=−11+574, −11−574-2x^2 - 11x - 8 = 0 \Rightarrow x = \frac{-11 + \sqrt{57}}{4},\ \frac{-11 - \sqrt{57}}{4}
Discriminant D = 57 · x ≈ -4.63746, -0.862541

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

Explain it step by step in chatMake a short video of it
Graph of y = -2x² - 11x - 8-7-6-5-4-3-2-11-30-20-1010x = -4.6374586x = -0.86254139vertexy = -2x² - 11x - 8
Graph of y = -2x² - 11x - 8. The curve crosses the x-axis at x = -4.6374586 and x = -0.86254139; the vertex is at (-2.75, 7.125).

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 = -8.

    −2x2−11x−8=0-2x^2 - 11x - 8 = 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)⋅(−8)=121−64=57D = (-11)^2 - 4\cdot(-2)\cdot(-8) = 121 - 64 = 57
  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±57−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{57}}{-4}
  5. 5

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

    x=−11+574orx=−11−574x = \frac{-11 + \sqrt{57}}{4}\quad\text{or}\quad x = \frac{-11 - \sqrt{57}}{4}
  6. 6

    As decimals: x ≈ -4.6374586 and x ≈ -0.86254139.

  7. 7

    Check numerically: for x ≈ -4.6374586 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 11x - 8 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{57}}{4},\ \frac{-11 - \sqrt{57}}{4} \]
\[ -2x^2 - 11x - 8 = 0 \]
\[ D = (-11)^2 - 4\cdot(-2)\cdot(-8) = 121 - 64 = 57 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{57}}{-4} \]
\[ x = \frac{-11 + \sqrt{57}}{4}\quad\text{or}\quad x = \frac{-11 - \sqrt{57}}{4} \]

The parabola y = -2x² - 11x - 8

xy
-5.75-10.875
-4.75-0.875
-3.755.125
-2.757.125
-1.755.125
-0.75-0.875
0.25-10.875

Questions

What are the solutions of -2x² - 11x - 8 = 0?

The solutions of -2x² - 11x - 8 = 0 are x = (-11 + √57)/4 or x = (-11 - √57)/4, that is x ≈ -4.6374586 and x ≈ -0.86254139.

Does -2x² - 11x - 8 = 0 factor?

Not over the integers, because the discriminant 57 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 11x - 8?

(-11/4, 57/8). 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 = 4.

Similar equations

    Solve -2x² - 11x - 8 = 0: x = (-11 + √57)/4 or x = (-11 - √57)/4 (steps) | WordGPT