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Solve -2x² + 3x + 5 = 0

The solutions of -2x² + 3x + 5 = 0 are x = -1 or x = 5/2.

x² +x += 0
−2x2+3x+5=0⇒x=−1, 52-2x^2 + 3x + 5 = 0 \Rightarrow x = -1,\ \frac{5}{2}
Discriminant D = 49 · x ≈ -1, 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 + 3x + 5-3-2-112345-30-25-20-15-10-5510x = -1x = 2.5vertexy = -2x² + 3x + 5
Graph of y = -2x² + 3x + 5. The curve crosses the x-axis at x = -1 and x = 2.5; the vertex is at (0.75, 6.125).

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 = 5.

    −2x2+3x+5=0-2x^2 + 3x + 5 = 0
  2. 2

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

    D=32−4⋅(−2)⋅5=9+40=49D = 3^2 - 4\cdot(-2)\cdot5 = 9 + 40 = 49
  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+1)(2x−5)=0-\left(x + 1\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+1=0⇒x=−12x−5=0⇒x=52x + 1 = 0 \Rightarrow x = -1\qquad 2x - 5 = 0 \Rightarrow x = \frac{5}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−3±49−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{49}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−1orx=52x = -1\quad\text{or}\quad x = \frac{5}{2}
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -2·(1) + 3·(-1) + 5 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 3x + 5 = 0\ \Rightarrow\ x = -1,\ \frac{5}{2} \]
\[ -2x^2 + 3x + 5 = 0 \]
\[ D = 3^2 - 4\cdot(-2)\cdot5 = 9 + 40 = 49 \]
\[ -\left(x + 1\right)\left(2x - 5\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 2x - 5 = 0 \Rightarrow x = \frac{5}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{49}}{-4} \]
\[ x = -1\quad\text{or}\quad x = \frac{5}{2} \]

The parabola y = -2x² + 3x + 5

xy
-2.25-11.875
-1.25-1.875
-0.254.125
0.756.125
1.754.125
2.75-1.875
3.75-11.875

Questions

What are the solutions of -2x² + 3x + 5 = 0?

The solutions of -2x² + 3x + 5 = 0 are x = -1 or x = 5/2.

Does -2x² + 3x + 5 = 0 factor?

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

What is the vertex of y = -2x² + 3x + 5?

(3/4, 49/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 3/2, product = c/a = -5/2.

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