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Solve 3x² - 11x + 10 = 0

The solutions of 3x² - 11x + 10 = 0 are x = 2 or x = 5/3.

x² +x += 0
3x2−11x+10=0⇒x=2, 533x^2 - 11x + 10 = 0 \Rightarrow x = 2,\ \frac{5}{3}
Discriminant D = 1 · x ≈ 2, 1.66667

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

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Graph of y = 3x² - 11x + 1012345101520x = 2x = 1.6666667vertexy = 3x² - 11x + 10
Graph of y = 3x² - 11x + 10. The curve crosses the x-axis at x = 2 and x = 1.6666667; the vertex is at (1.8333333, -0.083333333).

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 = 10.

    3x2−11x+10=03x^2 - 11x + 10 = 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⋅10=121−120=1D = (-11)^2 - 4\cdot3\cdot10 = 121 - 120 = 1
  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−5)=0\left(x - 2\right)\left(3x - 5\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−5=0⇒x=53x - 2 = 0 \Rightarrow x = 2\qquad 3x - 5 = 0 \Rightarrow x = \frac{5}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±16x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{1}}{6}
  7. 7

    Simplify to get the solutions:

    x=2orx=53x = 2\quad\text{or}\quad x = \frac{5}{3}
  8. 8

    Check by putting x = 2 back in: a·x² + b·x + c = 3·(4) + (-11)·(2) + 10 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 11x + 10 = 0\ \Rightarrow\ x = 2,\ \frac{5}{3} \]
\[ 3x^2 - 11x + 10 = 0 \]
\[ D = (-11)^2 - 4\cdot3\cdot10 = 121 - 120 = 1 \]
\[ \left(x - 2\right)\left(3x - 5\right) = 0 \]
\[ x - 2 = 0 \Rightarrow x = 2\qquad 3x - 5 = 0 \Rightarrow x = \frac{5}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{1}}{6} \]
\[ x = 2\quad\text{or}\quad x = \frac{5}{3} \]

The parabola y = 3x² - 11x + 10

xy
-1.166726.917267
-0.166711.917067
0.83332.9168667
1.8333-0.08333333
2.83332.9164667
3.833311.916267
4.833326.916067

Questions

What are the solutions of 3x² - 11x + 10 = 0?

The solutions of 3x² - 11x + 10 = 0 are x = 2 or x = 5/3.

Does 3x² - 11x + 10 = 0 factor?

Yes: (x-2)(3x-5) = 0.

What is the vertex of y = 3x² - 11x + 10?

(11/6, -1/12). 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 = 10/3.

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