WordGPT

Solve -2x² - 10x + 1 = 0

The solutions of -2x² - 10x + 1 = 0 are x = (-5 + 3√3)/2 or x = (-5 - 3√3)/2, that is x ≈ -5.0980762 and x ≈ 0.098076211.

x² +x += 0
−2x2−10x+1=0⇒x=−5+332, −5−332-2x^2 - 10x + 1 = 0 \Rightarrow x = \frac{-5 + 3\sqrt{3}}{2},\ \frac{-5 - 3\sqrt{3}}{2}
Discriminant D = 108 · x ≈ -5.09808, 0.0980762

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 + 1-7-6-5-4-3-2-112-40-30-20-101020x = -5.0980762x = 0.098076211vertexy = -2x² - 10x + 1
Graph of y = -2x² - 10x + 1. The curve crosses the x-axis at x = -5.0980762 and x = 0.098076211; the vertex is at (-2.5, 13.5).

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 = 1.

    −2x2−10x+1=0-2x^2 - 10x + 1 = 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)⋅1=100+8=108D = (-10)^2 - 4\cdot(-2)\cdot1 = 100 + 8 = 108
  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±108−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{108}}{-4}
  5. 5

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

    x=−5+332orx=−5−332x = \frac{-5 + 3\sqrt{3}}{2}\quad\text{or}\quad x = \frac{-5 - 3\sqrt{3}}{2}
  6. 6

    As decimals: x ≈ -5.0980762 and x ≈ 0.098076211.

  7. 7

    Check numerically: for x ≈ -5.0980762 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 - 10x + 1 = 0\ \Rightarrow\ x = \frac{-5 + 3\sqrt{3}}{2},\ \frac{-5 - 3\sqrt{3}}{2} \]
\[ -2x^2 - 10x + 1 = 0 \]
\[ D = (-10)^2 - 4\cdot(-2)\cdot1 = 100 + 8 = 108 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{10 \pm \sqrt{108}}{-4} \]
\[ x = \frac{-5 + 3\sqrt{3}}{2}\quad\text{or}\quad x = \frac{-5 - 3\sqrt{3}}{2} \]

The parabola y = -2x² - 10x + 1

xy
-5.5-4.5
-4.55.5
-3.511.5
-2.513.5
-1.511.5
-0.55.5
0.5-4.5

Questions

What are the solutions of -2x² - 10x + 1 = 0?

The solutions of -2x² - 10x + 1 = 0 are x = (-5 + 3√3)/2 or x = (-5 - 3√3)/2, that is x ≈ -5.0980762 and x ≈ 0.098076211.

Does -2x² - 10x + 1 = 0 factor?

Not over the integers, because the discriminant 108 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² - 10x + 1?

(-5/2, 27/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -5, product = c/a = -1/2.

Similar equations

    Solve -2x² - 10x + 1 = 0: x = (-5 + 3√3)/2 or x = (-5 - 3√3)/2 (steps) | WordGPT