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Solve -2x² + 9x - 10 = 0

The solutions of -2x² + 9x - 10 = 0 are x = 2 or x = 5/2.

x² +x += 0
−2x2+9x−10=0⇒x=2, 52-2x^2 + 9x - 10 = 0 \Rightarrow x = 2,\ \frac{5}{2}
Discriminant D = 1 · x ≈ 2, 2.5

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

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Graph of y = -2x² + 9x - 1012345-16-14-12-10-8-6-4-22x = 2x = 2.5vertexy = -2x² + 9x - 10
Graph of y = -2x² + 9x - 10. The curve crosses the x-axis at x = 2 and x = 2.5; the vertex is at (2.25, 0.125).

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 = -10.

    −2x2+9x−10=0-2x^2 + 9x - 10 = 0
  2. 2

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

    D=92−4⋅(−2)⋅(−10)=81−80=1D = 9^2 - 4\cdot(-2)\cdot(-10) = 81 - 80 = 1
  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:

    −(x−2)(2x−5)=0-\left(x - 2\right)\left(2x - 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:

    x−2=0⇒x=22x−5=0⇒x=52x - 2 = 0 \Rightarrow x = 2\qquad 2x - 5 = 0 \Rightarrow x = \frac{5}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−9±1−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{1}}{-4}
  7. 7

    Simplify to get the solutions:

    x=2orx=52x = 2\quad\text{or}\quad x = \frac{5}{2}
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = -2·(4) + 9·(2) + (-10) = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 9x - 10 = 0\ \Rightarrow\ x = 2,\ \frac{5}{2} \]
\[ -2x^2 + 9x - 10 = 0 \]
\[ D = 9^2 - 4\cdot(-2)\cdot(-10) = 81 - 80 = 1 \]
\[ -\left(x - 2\right)\left(2x - 5\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 2x - 5 = 0 \Rightarrow x = \frac{5}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-9 \pm \sqrt{1}}{-4} \]
\[ x = 2\quad\text{or}\quad x = \frac{5}{2} \]

The parabola y = -2x² + 9x - 10

xy
-0.75-17.875
0.25-7.875
1.25-1.875
2.250.125
3.25-1.875
4.25-7.875
5.25-17.875

Questions

What are the solutions of -2x² + 9x - 10 = 0?

The solutions of -2x² + 9x - 10 = 0 are x = 2 or x = 5/2.

Does -2x² + 9x - 10 = 0 factor?

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

What is the vertex of y = -2x² + 9x - 10?

(9/4, 1/8). 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.

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