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Solve 3x² - 8x - 11 = 0

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

x² +x += 0
3x2−8x−11=0⇒x=113, −13x^2 - 8x - 11 = 0 \Rightarrow x = \frac{11}{3},\ -1
Discriminant D = 196 · x ≈ 3.66667, -1

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

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Graph of y = 3x² - 8x - 11-3-2-1123456-20-101020304050x = 3.6666667x = -1vertexy = 3x² - 8x - 11
Graph of y = 3x² - 8x - 11. The curve crosses the x-axis at x = 3.6666667 and x = -1; 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=03x^2 - 8x - 11 = 0
  2. 2

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

    D=(−8)2−4⋅3⋅(−11)=64+132=196D = (-8)^2 - 4\cdot3\cdot(-11) = 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:

    (3x−11)(x+1)=0\left(3x - 11\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:

    3x−11=0⇒x=113x+1=0⇒x=−13x - 11 = 0 \Rightarrow x = \frac{11}{3}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=8±1966x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{196}}{6}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 11/3 back in: a·x² + b·x + c = 3·(121/9) + (-8)·(11/3) + (-11) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 8x - 11 = 0\ \Rightarrow\ x = \frac{11}{3},\ -1 \]
\[ 3x^2 - 8x - 11 = 0 \]
\[ D = (-8)^2 - 4\cdot3\cdot(-11) = 64 + 132 = 196 \]
\[ \left(3x - 11\right)\left(x + 1\right) = 0 \]
\[ 3x - 11 = 0 \Rightarrow x = \frac{11}{3}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{8 \pm \sqrt{196}}{6} \]
\[ x = \frac{11}{3}\quad\text{or}\quad x = -1 \]

The parabola y = 3x² - 8x - 11

xy
-1.666710.667267
-0.6667-4.3329333
0.3333-13.333133
1.3333-16.333333
2.3333-13.333533
3.3333-4.3337333
4.333310.666067

Questions

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

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

Does 3x² - 8x - 11 = 0 factor?

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

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

(4/3, -49/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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