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Solve -2x² + x + 3 = 0

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

x² +x += 0
−2x2+x+3=0⇒x=−1, 32-2x^2 + x + 3 = 0 \Rightarrow x = -1,\ \frac{3}{2}
Discriminant D = 25 · x ≈ -1, 1.5

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

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Graph of y = -2x² + x + 3-3-2-11234-25-20-15-10-55x = -1x = 1.5vertexy = -2x² + x + 3
Graph of y = -2x² + x + 3. The curve crosses the x-axis at x = -1 and x = 1.5; the vertex is at (0.25, 3.125).

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 = 1, c = 3.

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

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

    D=12−4⋅(−2)⋅3=1+24=25D = 1^2 - 4\cdot(-2)\cdot3 = 1 + 24 = 25
  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−3)=0-\left(x + 1\right)\left(2x - 3\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−3=0⇒x=32x + 1 = 0 \Rightarrow x = -1\qquad 2x - 3 = 0 \Rightarrow x = \frac{3}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−1±25−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{25}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−1orx=32x = -1\quad\text{or}\quad x = \frac{3}{2}
  8. 8

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

LaTeX source of the formulas on this page
\[ -2x^2 + x + 3 = 0\ \Rightarrow\ x = -1,\ \frac{3}{2} \]
\[ -2x^2 + x + 3 = 0 \]
\[ D = 1^2 - 4\cdot(-2)\cdot3 = 1 + 24 = 25 \]
\[ -\left(x + 1\right)\left(2x - 3\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 2x - 3 = 0 \Rightarrow x = \frac{3}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{25}}{-4} \]
\[ x = -1\quad\text{or}\quad x = \frac{3}{2} \]

The parabola y = -2x² + x + 3

xy
-2.75-14.875
-1.75-4.875
-0.751.125
0.253.125
1.251.125
2.25-4.875
3.25-14.875

Questions

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

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

Does -2x² + x + 3 = 0 factor?

Yes: -1(x+1)(2x-3) = 0.

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

(1/4, 25/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 1/2, product = c/a = -3/2.

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