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Solve -2x² + 8x + 9 = 0

The solutions of -2x² + 8x + 9 = 0 are x = (4 + √34)/2 or x = (4 - √34)/2, that is x ≈ -0.91547595 and x ≈ 4.9154759.

x² +x += 0
−2x2+8x+9=0⇒x=4+342, 4−342-2x^2 + 8x + 9 = 0 \Rightarrow x = \frac{4 + \sqrt{34}}{2},\ \frac{4 - \sqrt{34}}{2}
Discriminant D = 136 · x ≈ -0.915476, 4.91548

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

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Graph of y = -2x² + 8x + 9-2246-40-30-20-101020x = -0.91547595x = 4.9154759vertexy = -2x² + 8x + 9
Graph of y = -2x² + 8x + 9. The curve crosses the x-axis at x = -0.91547595 and x = 4.9154759; the vertex is at (2, 17).

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 = 8, c = 9.

    −2x2+8x+9=0-2x^2 + 8x + 9 = 0
  2. 2

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

    D=82−4⋅(−2)⋅9=64+72=136D = 8^2 - 4\cdot(-2)\cdot9 = 64 + 72 = 136
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−8±136−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{136}}{-4}
  5. 5

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

    x=4+342orx=4−342x = \frac{4 + \sqrt{34}}{2}\quad\text{or}\quad x = \frac{4 - \sqrt{34}}{2}
  6. 6

    As decimals: x ≈ -0.91547595 and x ≈ 4.9154759.

  7. 7

    Check numerically: for x ≈ -0.91547595 the left side is -1.7763568e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 8x + 9 = 0\ \Rightarrow\ x = \frac{4 + \sqrt{34}}{2},\ \frac{4 - \sqrt{34}}{2} \]
\[ -2x^2 + 8x + 9 = 0 \]
\[ D = 8^2 - 4\cdot(-2)\cdot9 = 64 + 72 = 136 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{136}}{-4} \]
\[ x = \frac{4 + \sqrt{34}}{2}\quad\text{or}\quad x = \frac{4 - \sqrt{34}}{2} \]

The parabola y = -2x² + 8x + 9

xy
-1-1
09
115
217
315
49
5-1

Questions

What are the solutions of -2x² + 8x + 9 = 0?

The solutions of -2x² + 8x + 9 = 0 are x = (4 + √34)/2 or x = (4 - √34)/2, that is x ≈ -0.91547595 and x ≈ 4.9154759.

Does -2x² + 8x + 9 = 0 factor?

Not over the integers, because the discriminant 136 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² + 8x + 9?

(2, 17). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 4, product = c/a = -9/2.

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    Solve -2x² + 8x + 9 = 0: x = (4 + √34)/2 or x = (4 - √34)/2 (steps) | WordGPT