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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, 433x^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 + 8-11234510152025x = 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=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=4D = (-10)^2 - 4\cdot3\cdot8 = 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±46x = \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
\[ 3x^2 - 10x + 8 = 0\ \Rightarrow\ x = 2,\ \frac{4}{3} \]
\[ 3x^2 - 10x + 8 = 0 \]
\[ D = (-10)^2 - 4\cdot3\cdot8 = 100 - 96 = 4 \]
\[ \left(x - 2\right)\left(3x - 4\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 3x - 4 = 0 \Rightarrow x = \frac{4}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{4}}{6} \]
\[ x = 2\quad\text{or}\quad x = \frac{4}{3} \]

The parabola y = 3x² - 10x + 8

xy
-1.333326.666067
-0.333311.666267
0.66672.6664667
1.6667-0.33333333
2.66672.6668667
3.666711.667067
4.666726.667267

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

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

(5/3, -1/3). 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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