WordGPT

Solve 2x² - 10x - 10 = 0

The solutions of 2x² - 10x - 10 = 0 are x = (5 + 3√5)/2 or x = (5 - 3√5)/2, that is x ≈ 5.854102 and x ≈ -0.85410197.

x² +x += 0
2x2−10x−10=0⇒x=5+352, 5−3522x^2 - 10x - 10 = 0 \Rightarrow x = \frac{5 + 3\sqrt{5}}{2},\ \frac{5 - 3\sqrt{5}}{2}
Discriminant D = 180 · x ≈ 5.8541, -0.854102

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

Explain it step by step in chatMake a short video of it
Graph of y = 2x² - 10x - 10-22468-30-20-101020304050x = 5.854102x = -0.85410197vertexy = 2x² - 10x - 10
Graph of y = 2x² - 10x - 10. The curve crosses the x-axis at x = 5.854102 and x = -0.85410197; the vertex is at (2.5, -22.5).

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 = -10.

    2x2−10x−10=02x^2 - 10x - 10 = 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⋅(−10)=100+80=180D = (-10)^2 - 4\cdot2\cdot(-10) = 100 + 80 = 180
  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±1804x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{180}}{4}
  5. 5

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

    x=5+352orx=5−352x = \frac{5 + 3\sqrt{5}}{2}\quad\text{or}\quad x = \frac{5 - 3\sqrt{5}}{2}
  6. 6

    As decimals: x ≈ 5.854102 and x ≈ -0.85410197.

  7. 7

    Check numerically: for x ≈ 5.854102 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 10x - 10 = 0\ \Rightarrow\ x = \frac{5 + 3\sqrt{5}}{2},\ \frac{5 - 3\sqrt{5}}{2} \]
\[ 2x^2 - 10x - 10 = 0 \]
\[ D = (-10)^2 - 4\cdot2\cdot(-10) = 100 + 80 = 180 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{180}}{4} \]
\[ x = \frac{5 + 3\sqrt{5}}{2}\quad\text{or}\quad x = \frac{5 - 3\sqrt{5}}{2} \]

The parabola y = 2x² - 10x - 10

xy
-0.5-4.5
0.5-14.5
1.5-20.5
2.5-22.5
3.5-20.5
4.5-14.5
5.5-4.5

Questions

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

The solutions of 2x² - 10x - 10 = 0 are x = (5 + 3√5)/2 or x = (5 - 3√5)/2, that is x ≈ 5.854102 and x ≈ -0.85410197.

Does 2x² - 10x - 10 = 0 factor?

Not over the integers, because the discriminant 180 is not a perfect square. Use the quadratic formula.

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

(5/2, -45/2). 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 = -5.

Similar equations

    Solve 2x² - 10x - 10 = 0: x = (5 + 3√5)/2 or x = (5 - 3√5)/2 (steps) | WordGPT