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Solve 2x² - 2x - 10 = 0

The solutions of 2x² - 2x - 10 = 0 are x = (1 + √21)/2 or x = (1 - √21)/2, that is x ≈ 2.7912878 and x ≈ -1.7912878.

x² +x += 0
2x2−2x−10=0⇒x=1+212, 1−2122x^2 - 2x - 10 = 0 \Rightarrow x = \frac{1 + \sqrt{21}}{2},\ \frac{1 - \sqrt{21}}{2}
Discriminant D = 84 · x ≈ 2.79129, -1.79129

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

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Graph of y = 2x² - 2x - 10-4-3-2-112345-10102030x = 2.7912878x = -1.7912878vertexy = 2x² - 2x - 10
Graph of y = 2x² - 2x - 10. The curve crosses the x-axis at x = 2.7912878 and x = -1.7912878; the vertex is at (0.5, -10.5).

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 = -2, c = -10.

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

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

    D=(−2)2−4⋅2⋅(−10)=4+80=84D = (-2)^2 - 4\cdot2\cdot(-10) = 4 + 80 = 84
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=2±844x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{84}}{4}
  5. 5

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

    x=1+212orx=1−212x = \frac{1 + \sqrt{21}}{2}\quad\text{or}\quad x = \frac{1 - \sqrt{21}}{2}
  6. 6

    As decimals: x ≈ 2.7912878 and x ≈ -1.7912878.

  7. 7

    Check numerically: for x ≈ 2.7912878 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 2x - 10 = 0\ \Rightarrow\ x = \frac{1 + \sqrt{21}}{2},\ \frac{1 - \sqrt{21}}{2} \]
\[ 2x^2 - 2x - 10 = 0 \]
\[ D = (-2)^2 - 4\cdot2\cdot(-10) = 4 + 80 = 84 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{2 \pm \sqrt{84}}{4} \]
\[ x = \frac{1 + \sqrt{21}}{2}\quad\text{or}\quad x = \frac{1 - \sqrt{21}}{2} \]

The parabola y = 2x² - 2x - 10

xy
-2.57.5
-1.5-2.5
-0.5-8.5
0.5-10.5
1.5-8.5
2.5-2.5
3.57.5

Questions

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

The solutions of 2x² - 2x - 10 = 0 are x = (1 + √21)/2 or x = (1 - √21)/2, that is x ≈ 2.7912878 and x ≈ -1.7912878.

Does 2x² - 2x - 10 = 0 factor?

Not over the integers, because the discriminant 84 is not a perfect square. Use the quadratic formula.

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

(1/2, -21/2). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 1, product = c/a = -5.

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