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Solve -3x² - 11x - 8 = 0

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

x² +x += 0
−3x2−11x−8=0⇒x=−83, −1-3x^2 - 11x - 8 = 0 \Rightarrow x = -\frac{8}{3},\ -1
Discriminant D = 25 · x ≈ -2.66667, -1

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

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Graph of y = -3x² - 11x - 8-5-4-3-2-11-30-25-20-15-10-55x = -2.6666667x = -1vertexy = -3x² - 11x - 8
Graph of y = -3x² - 11x - 8. The curve crosses the x-axis at x = -2.6666667 and x = -1; the vertex is at (-1.8333333, 2.0833333).

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 = -11, c = -8.

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

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

    D=(−11)2−4⋅(−3)⋅(−8)=121−96=25D = (-11)^2 - 4\cdot(-3)\cdot(-8) = 121 - 96 = 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:

    −(3x+8)(x+1)=0-\left(3x + 8\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+8=0⇒x=−83x+1=0⇒x=−13x + 8 = 0 \Rightarrow x = -\frac{8}{3}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±25−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{25}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−83orx=−1x = -\frac{8}{3}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = -8/3 back in: a·x² + b·x + c = -3·(64/9) + (-11)·(-8/3) + (-8) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 11x - 8 = 0\ \Rightarrow\ x = -\frac{8}{3},\ -1 \]
\[ -3x^2 - 11x - 8 = 0 \]
\[ D = (-11)^2 - 4\cdot(-3)\cdot(-8) = 121 - 96 = 25 \]
\[ -\left(3x + 8\right)\left(x + 1\right) = 0 \]
\[ 3x + 8 = 0 \Rightarrow x = -\frac{8}{3}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{25}}{-6} \]
\[ x = -\frac{8}{3}\quad\text{or}\quad x = -1 \]

The parabola y = -3x² - 11x - 8

xy
-4.8333-24.916067
-3.8333-9.9162667
-2.8333-0.91646667
-1.83332.0833333
-0.8333-0.91686667
0.1667-9.9170667
1.1667-24.917267

Questions

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

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

Does -3x² - 11x - 8 = 0 factor?

Yes: -1(3x+8)(x+1) = 0.

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

(-11/6, 25/12). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

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

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