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Solve -2x² + 11x + 6 = 0

The solutions of -2x² + 11x + 6 = 0 are x = -1/2 or x = 6.

x² +x += 0
−2x2+11x+6=0⇒x=−12, 6-2x^2 + 11x + 6 = 0 \Rightarrow x = -\frac{1}{2},\ 6
Discriminant D = 169 · x ≈ -0.5, 6

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

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Graph of y = -2x² + 11x + 6-22468-40-30-20-101020x = -0.5x = 6vertexy = -2x² + 11x + 6
Graph of y = -2x² + 11x + 6. The curve crosses the x-axis at x = -0.5 and x = 6; the vertex is at (2.75, 21.125).

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 = 11, c = 6.

    −2x2+11x+6=0-2x^2 + 11x + 6 = 0
  2. 2

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

    D=112−4⋅(−2)⋅6=121+48=169D = 11^2 - 4\cdot(-2)\cdot6 = 121 + 48 = 169
  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+1)(x−6)=0-\left(2x + 1\right)\left(x - 6\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+1=0⇒x=−12x−6=0⇒x=62x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad x - 6 = 0 \Rightarrow x = 6
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−11±169−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{169}}{-4}
  7. 7

    Simplify to get the solutions:

    x=−12orx=6x = -\frac{1}{2}\quad\text{or}\quad x = 6
  8. 8

    Check by putting x = -1/2 back in: a·x² + b·x + c = -2·(1/4) + 11·(-1/2) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 11x + 6 = 0\ \Rightarrow\ x = -\frac{1}{2},\ 6 \]
\[ -2x^2 + 11x + 6 = 0 \]
\[ D = 11^2 - 4\cdot(-2)\cdot6 = 121 + 48 = 169 \]
\[ -\left(2x + 1\right)\left(x - 6\right) = 0 \]
\[ 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad x - 6 = 0 \Rightarrow x = 6 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{169}}{-4} \]
\[ x = -\frac{1}{2}\quad\text{or}\quad x = 6 \]

The parabola y = -2x² + 11x + 6

xy
-0.253.125
0.7513.125
1.7519.125
2.7521.125
3.7519.125
4.7513.125
5.753.125

Questions

What are the solutions of -2x² + 11x + 6 = 0?

The solutions of -2x² + 11x + 6 = 0 are x = -1/2 or x = 6.

Does -2x² + 11x + 6 = 0 factor?

Yes: -1(2x+1)(x-6) = 0.

What is the vertex of y = -2x² + 11x + 6?

(11/4, 169/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 11/2, product = c/a = -3.

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