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Solve -2x² - 3x + 5 = 0

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

x² +x += 0
−2x2−3x+5=0⇒x=−52, 1-2x^2 - 3x + 5 = 0 \Rightarrow x = -\frac{5}{2},\ 1
Discriminant D = 49 · x ≈ -2.5, 1

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

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Graph of y = -2x² - 3x + 5-5-4-3-2-1123-30-25-20-15-10-5510x = -2.5x = 1vertexy = -2x² - 3x + 5
Graph of y = -2x² - 3x + 5. The curve crosses the x-axis at x = -2.5 and x = 1; 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=(−3)2−4⋅(−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:

    −(2x+5)(x−1)=0-\left(2x + 5\right)\left(x - 1\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+5=0⇒x=−52x−1=0⇒x=12x + 5 = 0 \Rightarrow x = -\frac{5}{2}\qquad x - 1 = 0 \Rightarrow x = 1
  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=−52orx=1x = -\frac{5}{2}\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -5/2 back in: a·x² + b·x + c = -2·(25/4) + (-3)·(-5/2) + 5 = 0 ✓

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

The parabola y = -2x² - 3x + 5

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

Questions

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

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

Does -2x² - 3x + 5 = 0 factor?

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