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Solve 3x² - 5x - 8 = 0

The solutions of 3x² - 5x - 8 = 0 are x = 8/3 or x = -1.

x² +x += 0
3x2−5x−8=0⇒x=83, −13x^2 - 5x - 8 = 0 \Rightarrow x = \frac{8}{3},\ -1
Discriminant D = 121 · 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 - 5x - 8-3-2-112345-101020304050x = 2.6666667x = -1vertexy = 3x² - 5x - 8
Graph of y = 3x² - 5x - 8. The curve crosses the x-axis at x = 2.6666667 and x = -1; the vertex is at (0.83333333, -10.083333).

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 = -5, c = -8.

    3x2−5x−8=03x^2 - 5x - 8 = 0
  2. 2

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

    D=(−5)2−4⋅3⋅(−8)=25+96=121D = (-5)^2 - 4\cdot3\cdot(-8) = 25 + 96 = 121
  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=5±1216x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{121}}{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) + (-5)·(8/3) + (-8) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 5x - 8 = 0\ \Rightarrow\ x = \frac{8}{3},\ -1 \]
\[ 3x^2 - 5x - 8 = 0 \]
\[ D = (-5)^2 - 4\cdot3\cdot(-8) = 25 + 96 = 121 \]
\[ \left(3x - 8\right)\left(x + 1\right) = 0 \]
\[ 3x - 8 = 0 \Rightarrow x = \frac{8}{3}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{5 \pm \sqrt{121}}{6} \]
\[ x = \frac{8}{3}\quad\text{or}\quad x = -1 \]

The parabola y = 3x² - 5x - 8

xy
-2.166716.917267
-1.16671.9170667
-0.1667-7.0831333
0.8333-10.083333
1.8333-7.0835333
2.83331.9162667
3.833316.916067

Questions

What are the solutions of 3x² - 5x - 8 = 0?

The solutions of 3x² - 5x - 8 = 0 are x = 8/3 or x = -1.

Does 3x² - 5x - 8 = 0 factor?

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

What is the vertex of y = 3x² - 5x - 8?

(5/6, -121/12). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5/3, product = c/a = -8/3.

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