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Solve 2x² - 10x - 7 = 0

The solutions of 2x² - 10x - 7 = 0 are x = (5 + √39)/2 or x = (5 - √39)/2, that is x ≈ 5.622499 and x ≈ -0.622499.

x² +x += 0
2x2−10x−7=0⇒x=5+392, 5−3922x^2 - 10x - 7 = 0 \Rightarrow x = \frac{5 + \sqrt{39}}{2},\ \frac{5 - \sqrt{39}}{2}
Discriminant D = 156 · x ≈ 5.6225, -0.622499

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

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Graph of y = 2x² - 10x - 7-22468-20-1010203040x = 5.622499x = -0.622499vertexy = 2x² - 10x - 7
Graph of y = 2x² - 10x - 7. The curve crosses the x-axis at x = 5.622499 and x = -0.622499; the vertex is at (2.5, -19.5).

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 = -7.

    2x2−10x−7=02x^2 - 10x - 7 = 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⋅(−7)=100+56=156D = (-10)^2 - 4\cdot2\cdot(-7) = 100 + 56 = 156
  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±1564x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{156}}{4}
  5. 5

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

    x=5+392orx=5−392x = \frac{5 + \sqrt{39}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{39}}{2}
  6. 6

    As decimals: x ≈ 5.622499 and x ≈ -0.622499.

  7. 7

    Check numerically: for x ≈ 5.622499 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 10x - 7 = 0\ \Rightarrow\ x = \frac{5 + \sqrt{39}}{2},\ \frac{5 - \sqrt{39}}{2} \]
\[ 2x^2 - 10x - 7 = 0 \]
\[ D = (-10)^2 - 4\cdot2\cdot(-7) = 100 + 56 = 156 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{156}}{4} \]
\[ x = \frac{5 + \sqrt{39}}{2}\quad\text{or}\quad x = \frac{5 - \sqrt{39}}{2} \]

The parabola y = 2x² - 10x - 7

xy
-0.5-1.5
0.5-11.5
1.5-17.5
2.5-19.5
3.5-17.5
4.5-11.5
5.5-1.5

Questions

What are the solutions of 2x² - 10x - 7 = 0?

The solutions of 2x² - 10x - 7 = 0 are x = (5 + √39)/2 or x = (5 - √39)/2, that is x ≈ 5.622499 and x ≈ -0.622499.

Does 2x² - 10x - 7 = 0 factor?

Not over the integers, because the discriminant 156 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 10x - 7?

(5/2, -39/2). 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 = -7/2.

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    Solve 2x² - 10x - 7 = 0: x = (5 + √39)/2 or x = (5 - √39)/2 (steps) | WordGPT