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Solve 3x² - 11x + 6 = 0

The solutions of 3x² - 11x + 6 = 0 are x = 3 or x = 2/3.

x² +x += 0
3x2−11x+6=0⇒x=3, 233x^2 - 11x + 6 = 0 \Rightarrow x = 3,\ \frac{2}{3}
Discriminant D = 49 · x ≈ 3, 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 - 11x + 6-112345102030x = 3x = 0.66666667vertexy = 3x² - 11x + 6
Graph of y = 3x² - 11x + 6. The curve crosses the x-axis at x = 3 and x = 0.66666667; the vertex is at (1.8333333, -4.0833333).

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 = 6.

    3x2−11x+6=03x^2 - 11x + 6 = 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⋅6=121−72=49D = (-11)^2 - 4\cdot3\cdot6 = 121 - 72 = 49
  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−3)(3x−2)=0\left(x - 3\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−3=0⇒x=33x−2=0⇒x=23x - 3 = 0 \Rightarrow x = 3\qquad 3x - 2 = 0 \Rightarrow x = \frac{2}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±496x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{49}}{6}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = 3 back in: a·x² + b·x + c = 3·(9) + (-11)·(3) + 6 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 11x + 6 = 0\ \Rightarrow\ x = 3,\ \frac{2}{3} \]
\[ 3x^2 - 11x + 6 = 0 \]
\[ D = (-11)^2 - 4\cdot3\cdot6 = 121 - 72 = 49 \]
\[ \left(x - 3\right)\left(3x - 2\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 3x - 2 = 0 \Rightarrow x = \frac{2}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{49}}{6} \]
\[ x = 3\quad\text{or}\quad x = \frac{2}{3} \]

The parabola y = 3x² - 11x + 6

xy
-1.166722.917267
-0.16677.9170667
0.8333-1.0831333
1.8333-4.0833333
2.8333-1.0835333
3.83337.9162667
4.833322.916067

Questions

What are the solutions of 3x² - 11x + 6 = 0?

The solutions of 3x² - 11x + 6 = 0 are x = 3 or x = 2/3.

Does 3x² - 11x + 6 = 0 factor?

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

What is the vertex of y = 3x² - 11x + 6?

(11/6, -49/12). 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 = 2.

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