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Solve -2x² + x + 6 = 0

The solutions of -2x² + x + 6 = 0 are x = -3/2 or x = 2.

x² +x += 0
−2x2+x+6=0⇒x=−32, 2-2x^2 + x + 6 = 0 \Rightarrow x = -\frac{3}{2},\ 2
Discriminant D = 49 · x ≈ -1.5, 2

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

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Graph of y = -2x² + x + 6-4-3-2-11234-30-25-20-15-10-5510x = -1.5x = 2vertexy = -2x² + x + 6
Graph of y = -2x² + x + 6. The curve crosses the x-axis at x = -1.5 and x = 2; the vertex is at (0.25, 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 = 1, c = 6.

    −2x2+x+6=0-2x^2 + x + 6 = 0
  2. 2

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

    D=12−4⋅(−2)⋅6=1+48=49D = 1^2 - 4\cdot(-2)\cdot6 = 1 + 48 = 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+3)(x−2)=0-\left(2x + 3\right)\left(x - 2\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+3=0⇒x=−32x−2=0⇒x=22x + 3 = 0 \Rightarrow x = -\frac{3}{2}\qquad x - 2 = 0 \Rightarrow x = 2
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−1±49−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{49}}{-4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -3/2 back in: a·x² + b·x + c = -2·(9/4) + 1·(-3/2) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + x + 6 = 0\ \Rightarrow\ x = -\frac{3}{2},\ 2 \]
\[ -2x^2 + x + 6 = 0 \]
\[ D = 1^2 - 4\cdot(-2)\cdot6 = 1 + 48 = 49 \]
\[ -\left(2x + 3\right)\left(x - 2\right) = 0 \]
\[ 2x + 3 = 0 \Rightarrow x = -\frac{3}{2}\qquad x - 2 = 0 \Rightarrow x = 2 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-1 \pm \sqrt{49}}{-4} \]
\[ x = -\frac{3}{2}\quad\text{or}\quad x = 2 \]

The parabola y = -2x² + x + 6

xy
-2.75-11.875
-1.75-1.875
-0.754.125
0.256.125
1.254.125
2.25-1.875
3.25-11.875

Questions

What are the solutions of -2x² + x + 6 = 0?

The solutions of -2x² + x + 6 = 0 are x = -3/2 or x = 2.

Does -2x² + x + 6 = 0 factor?

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

What is the vertex of y = -2x² + x + 6?

(1/4, 49/8). 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.

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