WordGPT

Solve -x² - 3x - 11 = 0

-x² - 3x - 11 = 0 has no real solutions. The two complex solutions are x = (-3 + √35 i)/2 and x = (-3 - √35 i)/2.

x² +x += 0
−x2−3x−11=0⇒x=−3+35 i2, −3−35 i2 (complex)-x^2 - 3x - 11 = 0 \Rightarrow x = \frac{-3 + \sqrt{35}\, i}{2},\ \frac{-3 - \sqrt{35}\, i}{2}\ \text{(complex)}
Discriminant D = -35

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 = -x² - 3x - 11-5-4-3-2-112-24-22-20-18-16-14-12-10-8vertexy = -x² - 3x - 11
Graph of y = -x² - 3x - 11. The curve never reaches the x-axis, which is why there are no real solutions; the vertex is at (-1.5, -8.75).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = -3, c = -11.

    −x2−3x−11=0-x^2 - 3x - 11 = 0
  2. 2

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

    D=(−3)2−4⋅(−1)⋅(−11)=9−44=−35D = (-3)^2 - 4\cdot(-1)\cdot(-11) = 9 - 44 = -35
  3. 3

    D is negative, so there are no real solutions; the two solutions are complex numbers.

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=3±−35−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-35}}{-2}
  5. 5

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

    x=−3+35 i2orx=−3−35 i2x = \frac{-3 + \sqrt{35}\, i}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{35}\, i}{2}
LaTeX source of the formulas on this page
\[ -x^2 - 3x - 11 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{35}\, i}{2},\ \frac{-3 - \sqrt{35}\, i}{2} \]
\[ -x^2 - 3x - 11 = 0 \]
\[ D = (-3)^2 - 4\cdot(-1)\cdot(-11) = 9 - 44 = -35 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{3 \pm \sqrt{-35}}{-2} \]
\[ x = \frac{-3 + \sqrt{35}\, i}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{35}\, i}{2} \]

The parabola y = -x² - 3x - 11

xy
-4.5-17.75
-3.5-12.75
-2.5-9.75
-1.5-8.75
-0.5-9.75
0.5-12.75
1.5-17.75

Questions

What are the solutions of -x² - 3x - 11 = 0?

-x² - 3x - 11 = 0 has no real solutions. The two complex solutions are x = (-3 + √35 i)/2 and x = (-3 - √35 i)/2.

Does -x² - 3x - 11 = 0 factor?

Not over the integers, because the discriminant -35 is negative. Use the quadratic formula.

What is the vertex of y = -x² - 3x - 11?

(-3/2, -35/4). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = 11.

Similar equations

    Solve -x² - 3x - 11 = 0: no real solutions (steps) | WordGPT