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Solve -3x² - 10x - 8 = 0

The solutions of -3x² - 10x - 8 = 0 are x = -2 or x = -4/3.

x² +x += 0
−3x2−10x−8=0⇒x=−2, −43-3x^2 - 10x - 8 = 0 \Rightarrow x = -2,\ -\frac{4}{3}
Discriminant D = 4 · x ≈ -2, -1.33333

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

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Graph of y = -3x² - 10x - 8-4-3-2-11-25-20-15-10-5x = -2x = -1.3333333vertexy = -3x² - 10x - 8
Graph of y = -3x² - 10x - 8. The curve crosses the x-axis at x = -2 and x = -1.3333333; the vertex is at (-1.6666667, 0.33333333).

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 = -10, c = -8.

    −3x2−10x−8=0-3x^2 - 10x - 8 = 0
  2. 2

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

    D=(−10)2−4⋅(−3)⋅(−8)=100−96=4D = (-10)^2 - 4\cdot(-3)\cdot(-8) = 100 - 96 = 4
  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+2)(3x+4)=0-\left(x + 2\right)\left(3x + 4\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+2=0⇒x=−23x+4=0⇒x=−43x + 2 = 0 \Rightarrow x = -2\qquad 3x + 4 = 0 \Rightarrow x = -\frac{4}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±4−6x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{4}}{-6}
  7. 7

    Simplify to get the solutions:

    x=−2orx=−43x = -2\quad\text{or}\quad x = -\frac{4}{3}
  8. 8

    Check by putting x = -2 back in: a·x² + b·x + c = -3·(4) + (-10)·(-2) + (-8) = 0 ✓

LaTeX source of the formulas on this page
\[ -3x^2 - 10x - 8 = 0\ \Rightarrow\ x = -2,\ -\frac{4}{3} \]
\[ -3x^2 - 10x - 8 = 0 \]
\[ D = (-10)^2 - 4\cdot(-3)\cdot(-8) = 100 - 96 = 4 \]
\[ -\left(x + 2\right)\left(3x + 4\right) = 0 \]
\[ x + 2 = 0 \Rightarrow x = -2\qquad 3x + 4 = 0 \Rightarrow x = -\frac{4}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{4}}{-6} \]
\[ x = -2\quad\text{or}\quad x = -\frac{4}{3} \]

The parabola y = -3x² - 10x - 8

xy
-4.6667-26.667267
-3.6667-11.667067
-2.6667-2.6668667
-1.66670.33333333
-0.6667-2.6664667
0.3333-11.666267
1.3333-26.666067

Questions

What are the solutions of -3x² - 10x - 8 = 0?

The solutions of -3x² - 10x - 8 = 0 are x = -2 or x = -4/3.

Does -3x² - 10x - 8 = 0 factor?

Yes: -1(x+2)(3x+4) = 0.

What is the vertex of y = -3x² - 10x - 8?

(-5/3, 1/3). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -10/3, product = c/a = 8/3.

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