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Solve 3x² - 10x - 8 = 0

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

x² +x += 0
3x2−10x−8=0⇒x=4, −233x^2 - 10x - 8 = 0 \Rightarrow x = 4,\ -\frac{2}{3}
Discriminant D = 196 · x ≈ 4, -0.666667

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

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Graph of y = 3x² - 10x - 8-3-2-1123456-20-101020304050x = 4x = -0.66666667vertexy = 3x² - 10x - 8
Graph of y = 3x² - 10x - 8. The curve crosses the x-axis at x = 4 and x = -0.66666667; the vertex is at (1.6666667, -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 = -10, c = -8.

    3x2−10x−8=03x^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=196D = (-10)^2 - 4\cdot3\cdot(-8) = 100 + 96 = 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−4)(3x+2)=0\left(x - 4\right)\left(3x + 2\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−4=0⇒x=43x+2=0⇒x=−23x - 4 = 0 \Rightarrow x = 4\qquad 3x + 2 = 0 \Rightarrow x = -\frac{2}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±1966x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{196}}{6}
  7. 7

    Simplify to get the solutions:

    x=4orx=−23x = 4\quad\text{or}\quad x = -\frac{2}{3}
  8. 8

    Check by putting x = 4 back in: a·x² + b·x + c = 3·(16) + (-10)·(4) + (-8) = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 10x - 8 = 0\ \Rightarrow\ x = 4,\ -\frac{2}{3} \]
\[ 3x^2 - 10x - 8 = 0 \]
\[ D = (-10)^2 - 4\cdot3\cdot(-8) = 100 + 96 = 196 \]
\[ \left(x - 4\right)\left(3x + 2\right) = 0 \]
\[ x - 4 = 0 \Rightarrow x = 4\qquad 3x + 2 = 0 \Rightarrow x = -\frac{2}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{196}}{6} \]
\[ x = 4\quad\text{or}\quad x = -\frac{2}{3} \]

The parabola y = 3x² - 10x - 8

xy
-1.333310.666067
-0.3333-4.3337333
0.6667-13.333533
1.6667-16.333333
2.6667-13.333133
3.6667-4.3329333
4.666710.667267

Questions

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

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

Does 3x² - 10x - 8 = 0 factor?

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

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

(5/3, -49/3). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

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

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