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Solve 3x² - 10x + 3 = 0

The solutions of 3x² - 10x + 3 = 0 are x = 3 or x = 1/3.

x² +x += 0
3x2−10x+3=0⇒x=3, 133x^2 - 10x + 3 = 0 \Rightarrow x = 3,\ \frac{1}{3}
Discriminant D = 64 · x ≈ 3, 0.333333

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

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Graph of y = 3x² - 10x + 3-2-112345-1010203040x = 3x = 0.33333333vertexy = 3x² - 10x + 3
Graph of y = 3x² - 10x + 3. The curve crosses the x-axis at x = 3 and x = 0.33333333; the vertex is at (1.6666667, -5.3333333).

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 = 3.

    3x2−10x+3=03x^2 - 10x + 3 = 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⋅3=100−36=64D = (-10)^2 - 4\cdot3\cdot3 = 100 - 36 = 64
  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−1)=0\left(x - 3\right)\left(3x - 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:

    x−3=0⇒x=33x−1=0⇒x=13x - 3 = 0 \Rightarrow x = 3\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=10±646x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{64}}{6}
  7. 7

    Simplify to get the solutions:

    x=3orx=13x = 3\quad\text{or}\quad x = \frac{1}{3}
  8. 8

    Check by putting x = 3 back in: a·x² + b·x + c = 3·(9) + (-10)·(3) + 3 = 0 ✓

LaTeX source of the formulas on this page
\[ 3x^2 - 10x + 3 = 0\ \Rightarrow\ x = 3,\ \frac{1}{3} \]
\[ 3x^2 - 10x + 3 = 0 \]
\[ D = (-10)^2 - 4\cdot3\cdot3 = 100 - 36 = 64 \]
\[ \left(x - 3\right)\left(3x - 1\right) = 0 \]
\[ x - 3 = 0 \Rightarrow x = 3\qquad 3x - 1 = 0 \Rightarrow x = \frac{1}{3} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{64}}{6} \]
\[ x = 3\quad\text{or}\quad x = \frac{1}{3} \]

The parabola y = 3x² - 10x + 3

xy
-1.333321.666067
-0.33336.6662667
0.6667-2.3335333
1.6667-5.3333333
2.6667-2.3331333
3.66676.6670667
4.666721.667267

Questions

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

The solutions of 3x² - 10x + 3 = 0 are x = 3 or x = 1/3.

Does 3x² - 10x + 3 = 0 factor?

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

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

(5/3, -16/3). 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 = 1.

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