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Solve 2x² + 3x - 8 = 0

The solutions of 2x² + 3x - 8 = 0 are x = (-3 + √73)/4 or x = (-3 - √73)/4, that is x ≈ 1.3860009 and x ≈ -2.8860009.

x² +x += 0
2x2+3x−8=0⇒x=−3+734, −3−7342x^2 + 3x - 8 = 0 \Rightarrow x = \frac{-3 + \sqrt{73}}{4},\ \frac{-3 - \sqrt{73}}{4}
Discriminant D = 73 · x ≈ 1.386, -2.886

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

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Graph of y = 2x² + 3x - 8-5-4-3-2-1123-10102030x = 1.3860009x = -2.8860009vertexy = 2x² + 3x - 8
Graph of y = 2x² + 3x - 8. The curve crosses the x-axis at x = 1.3860009 and x = -2.8860009; the vertex is at (-0.75, -9.125).

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 = 3, c = -8.

    2x2+3x−8=02x^2 + 3x - 8 = 0
  2. 2

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

    D=32−4⋅2⋅(−8)=9+64=73D = 3^2 - 4\cdot2\cdot(-8) = 9 + 64 = 73
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−3±734x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{73}}{4}
  5. 5

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

    x=−3+734orx=−3−734x = \frac{-3 + \sqrt{73}}{4}\quad\text{or}\quad x = \frac{-3 - \sqrt{73}}{4}
  6. 6

    As decimals: x ≈ 1.3860009 and x ≈ -2.8860009.

  7. 7

    Check numerically: for x ≈ 1.3860009 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 + 3x - 8 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{73}}{4},\ \frac{-3 - \sqrt{73}}{4} \]
\[ 2x^2 + 3x - 8 = 0 \]
\[ D = 3^2 - 4\cdot2\cdot(-8) = 9 + 64 = 73 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{73}}{4} \]
\[ x = \frac{-3 + \sqrt{73}}{4}\quad\text{or}\quad x = \frac{-3 - \sqrt{73}}{4} \]

The parabola y = 2x² + 3x - 8

xy
-3.758.875
-2.75-1.125
-1.75-7.125
-0.75-9.125
0.25-7.125
1.25-1.125
2.258.875

Questions

What are the solutions of 2x² + 3x - 8 = 0?

The solutions of 2x² + 3x - 8 = 0 are x = (-3 + √73)/4 or x = (-3 - √73)/4, that is x ≈ 1.3860009 and x ≈ -2.8860009.

Does 2x² + 3x - 8 = 0 factor?

Not over the integers, because the discriminant 73 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² + 3x - 8?

(-3/4, -73/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3/2, product = c/a = -4.

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    Solve 2x² + 3x - 8 = 0: x = (-3 + √73)/4 or x = (-3 - √73)/4 (steps) | WordGPT