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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, 13x^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 + 8-112345-551015202530x = 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=03x^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\cdot3\cdot8 = 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±256x = \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
\[ 3x^2 - 11x + 8 = 0\ \Rightarrow\ x = \frac{8}{3},\ 1 \]
\[ 3x^2 - 11x + 8 = 0 \]
\[ D = (-11)^2 - 4\cdot3\cdot8 = 121 - 96 = 25 \]
\[ \left(3x - 8\right)\left(x - 1\right) = 0 \]
\[ 3x - 8 = 0 \Rightarrow x = \frac{8}{3}\qquad x - 1 = 0 \Rightarrow x = 1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{25}}{6} \]
\[ x = \frac{8}{3}\quad\text{or}\quad x = 1 \]

The parabola y = 3x² - 11x + 8

xy
-1.166724.917267
-0.16679.9170667
0.83330.91686667
1.8333-2.0833333
2.83330.91646667
3.83339.9162667
4.833324.916067

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

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

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

What are the sum and product of the roots?

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

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