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Solve 2x² + 11x - 8 = 0

The solutions of 2x² + 11x - 8 = 0 are x = (-11 + √185)/4 or x = (-11 - √185)/4, that is x ≈ 0.65036763 and x ≈ -6.1503676.

x² +x += 0
2x2+11x−8=0⇒x=−11+1854, −11−18542x^2 + 11x - 8 = 0 \Rightarrow x = \frac{-11 + \sqrt{185}}{4},\ \frac{-11 - \sqrt{185}}{4}
Discriminant D = 185 · x ≈ 0.650368, -6.15037

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

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Graph of y = 2x² + 11x - 8-8-6-4-22-30-20-101020304050x = 0.65036763x = -6.1503676vertexy = 2x² + 11x - 8
Graph of y = 2x² + 11x - 8. The curve crosses the x-axis at x = 0.65036763 and x = -6.1503676; the vertex is at (-2.75, -23.125).

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=02x^2 + 11x - 8 = 0
  2. 2

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

    D=112−4⋅2⋅(−8)=121+64=185D = 11^2 - 4\cdot2\cdot(-8) = 121 + 64 = 185
  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±1854x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{185}}{4}
  5. 5

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

    x=−11+1854orx=−11−1854x = \frac{-11 + \sqrt{185}}{4}\quad\text{or}\quad x = \frac{-11 - \sqrt{185}}{4}
  6. 6

    As decimals: x ≈ 0.65036763 and x ≈ -6.1503676.

  7. 7

    Check numerically: for x ≈ 0.65036763 the left side is 1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 11x - 8 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{185}}{4},\ \frac{-11 - \sqrt{185}}{4} \]
\[ 2x^2 + 11x - 8 = 0 \]
\[ D = 11^2 - 4\cdot2\cdot(-8) = 121 + 64 = 185 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{185}}{4} \]
\[ x = \frac{-11 + \sqrt{185}}{4}\quad\text{or}\quad x = \frac{-11 - \sqrt{185}}{4} \]

The parabola y = 2x² + 11x - 8

xy
-5.75-5.125
-4.75-15.125
-3.75-21.125
-2.75-23.125
-1.75-21.125
-0.75-15.125
0.25-5.125

Questions

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

The solutions of 2x² + 11x - 8 = 0 are x = (-11 + √185)/4 or x = (-11 - √185)/4, that is x ≈ 0.65036763 and x ≈ -6.1503676.

Does 2x² + 11x - 8 = 0 factor?

Not over the integers, because the discriminant 185 is not a perfect square. Use the quadratic formula.

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

(-11/4, -185/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11/2, product = c/a = -4.

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    Solve 2x² + 11x - 8 = 0: x = (-11 + √185)/4 or x = (-11 - √185)/4 (steps) | WordGPT