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Solve 2x² - 10x + 6 = 0

The solutions of 2x² - 10x + 6 = 0 are x = (5 + √13)/2 or x = (5 - √13)/2, that is x ≈ 4.3027756 and x ≈ 0.69722436.

x² +x += 0
2x2−10x+6=0⇒x=5+132, 5−1322x^2 - 10x + 6 = 0 \Rightarrow x = \frac{5 + \sqrt{13}}{2},\ \frac{5 - \sqrt{13}}{2}
Discriminant D = 52 · x ≈ 4.30278, 0.697224

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

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Graph of y = 2x² - 10x + 6-1123456-10102030x = 4.3027756x = 0.69722436vertexy = 2x² - 10x + 6
Graph of y = 2x² - 10x + 6. The curve crosses the x-axis at x = 4.3027756 and x = 0.69722436; the vertex is at (2.5, -6.5).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 2, b = -10, c = 6.

    2x2−10x+6=02x^2 - 10x + 6 = 0
  2. 2

    Work out the discriminant D = b² − 4ac. Its sign tells you how many real solutions there are.

    D=(−10)2−4⋅2⋅6=100−48=52D = (-10)^2 - 4\cdot2\cdot6 = 100 - 48 = 52
  3. 3

    D is positive, so there are two different real solutions.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=10±524x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{52}}{4}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=5+132orx=5−132x = \frac{5 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{13}}{2}
  6. 6

    As decimals: x ≈ 4.3027756 and x ≈ 0.69722436.

  7. 7

    Check numerically: for x ≈ 4.3027756 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 10x + 6 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{13}}{2},\ \frac{5 - \sqrt{13}}{2} \]
\[ 2x^2 - 10x + 6 = 0 \]
\[ D = (-10)^2 - 4\cdot2\cdot6 = 100 - 48 = 52 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{52}}{4} \]
\[ x = \frac{5 + \sqrt{13}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{13}}{2} \]

The parabola y = 2x² - 10x + 6

xy
-0.511.5
0.51.5
1.5-4.5
2.5-6.5
3.5-4.5
4.51.5
5.511.5

Questions

What are the solutions of 2x² - 10x + 6 = 0?

The solutions of 2x² - 10x + 6 = 0 are x = (5 + √13)/2 or x = (5 - √13)/2, that is x ≈ 4.3027756 and x ≈ 0.69722436.

Does 2x² - 10x + 6 = 0 factor?

Not over the integers, because the discriminant 52 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 10x + 6?

(5/2, -13/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 5, product = c/a = 3.

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