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Solve -3x² + 8x + 11 = 0

The solutions of -3x² + 8x + 11 = 0 are x = -1 or x = 11/3.

x² +x += 0
−3x2+8x+11=0⇒x=−1, 113-3x^2 + 8x + 11 = 0 \Rightarrow x = -1,\ \frac{11}{3}
Discriminant D = 196 · x ≈ -1, 3.66667

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

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Graph of y = -3x² + 8x + 11-3-2-1123456-50-40-30-20-101020x = -1x = 3.6666667vertexy = -3x² + 8x + 11
Graph of y = -3x² + 8x + 11. The curve crosses the x-axis at x = -1 and x = 3.6666667; the vertex is at (1.3333333, 16.333333).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -3, b = 8, c = 11.

    −3x2+8x+11=0-3x^2 + 8x + 11 = 0
  2. 2

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

    D=82−4⋅(−3)⋅11=64+132=196D = 8^2 - 4\cdot(-3)\cdot11 = 64 + 132 = 196
  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)(3x−11)=0-\left(x + 1\right)\left(3x - 11\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=−13x−11=0⇒x=113x + 1 = 0 \Rightarrow x = -1\qquad 3x - 11 = 0 \Rightarrow x = \frac{11}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=−8±196−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{196}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−1orx=113x = -1\quad\text{or}\quad x = \frac{11}{3}
  8. 8

    Check by putting x = -1 back in: a·x² + b·x + c = -3·(1) + 8·(-1) + 11 = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 + 8x + 11 = 0\ \Rightarrow\ x = -1,\ \frac{11}{3} \]
\[ -3x^2 + 8x + 11 = 0 \]
\[ D = 8^2 - 4\cdot(-3)\cdot11 = 64 + 132 = 196 \]
\[ -\left(x + 1\right)\left(3x - 11\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 3x - 11 = 0 \Rightarrow x = \frac{11}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-8 \pm \sqrt{196}}{-6} \]
\[ x = -1\quad\text{or}\quad x = \frac{11}{3} \]

The parabola y = -3x² + 8x + 11

xy
-1.6667-10.667267
-0.66674.3329333
0.333313.333133
1.333316.333333
2.333313.333533
3.33334.3337333
4.3333-10.666067

Questions

What are the solutions of -3x² + 8x + 11 = 0?

The solutions of -3x² + 8x + 11 = 0 are x = -1 or x = 11/3.

Does -3x² + 8x + 11 = 0 factor?

Yes: -1(x+1)(3x-11) = 0.

What is the vertex of y = -3x² + 8x + 11?

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

What are the sum and product of the roots?

Sum = −b/a = 8/3, product = c/a = -11/3.

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