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Solve -2x² + 9x + 5 = 0

The solutions of -2x² + 9x + 5 = 0 are x = -1/2 or x = 5.

x² +x += 0
−2x2+9x+5=0⇒x=−12, 5-2x^2 + 9x + 5 = 0 \Rightarrow x = -\frac{1}{2},\ 5
Discriminant D = 121 · x ≈ -0.5, 5

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

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Graph of y = -2x² + 9x + 5-2246-40-30-20-101020x = -0.5x = 5vertexy = -2x² + 9x + 5
Graph of y = -2x² + 9x + 5. The curve crosses the x-axis at x = -0.5 and x = 5; the vertex is at (2.25, 15.125).

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 = 9, c = 5.

    −2x2+9x+5=0-2x^2 + 9x + 5 = 0
  2. 2

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

    D=92−4⋅(−2)⋅5=81+40=121D = 9^2 - 4\cdot(-2)\cdot5 = 81 + 40 = 121
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    −(2x+1)(x−5)=0-\left(2x + 1\right)\left(x - 5\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    2x+1=0⇒x=−12x−5=0⇒x=52x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad x - 5 = 0 \Rightarrow x = 5
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−9±121−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{121}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−12orx=5x = -\frac{1}{2}\quad\text{or}\quad x = 5
  8. 8

    Check by putting x = -1/2 back in: a·x² + b·x + c = -2·(1/4) + 9·(-1/2) + 5 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 9x + 5 = 0\ \Rightarrow\ x = -\frac{1}{2},\ 5 \]
\[ -2x^2 + 9x + 5 = 0 \]
\[ D = 9^2 - 4\cdot(-2)\cdot5 = 81 + 40 = 121 \]
\[ -\left(2x + 1\right)\left(x - 5\right) = 0 \]
\[ 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad x - 5 = 0 \Rightarrow x = 5 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{121}}{-4} \]
\[ x = -\frac{1}{2}\quad\text{or}\quad x = 5 \]

The parabola y = -2x² + 9x + 5

xy
-0.75-2.875
0.257.125
1.2513.125
2.2515.125
3.2513.125
4.257.125
5.25-2.875

Questions

What are the solutions of -2x² + 9x + 5 = 0?

The solutions of -2x² + 9x + 5 = 0 are x = -1/2 or x = 5.

Does -2x² + 9x + 5 = 0 factor?

Yes: -1(2x+1)(x-5) = 0.

What is the vertex of y = -2x² + 9x + 5?

(9/4, 121/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 9/2, product = c/a = -5/2.

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