WordGPT

Solve x² + 11x + 2 = 0

The solutions of x² + 11x + 2 = 0 are x = (-11 + √113)/2 or x = (-11 - √113)/2, that is x ≈ -0.18492709 and x ≈ -10.815073.

x² +x += 0
x2+11x+2=0⇒x=−11+1132, −11−1132x^2 + 11x + 2 = 0 \Rightarrow x = \frac{-11 + \sqrt{113}}{2},\ \frac{-11 - \sqrt{113}}{2}
Discriminant D = 113 · x ≈ -0.184927, -10.8151

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 + 11x + 2-12-10-8-6-4-22-30-20-10102030x = -0.18492709x = -10.815073vertexy = x² + 11x + 2
Graph of y = x² + 11x + 2. The curve crosses the x-axis at x = -0.18492709 and x = -10.815073; the vertex is at (-5.5, -28.25).

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 = 11, c = 2.

    x2+11x+2=0x^2 + 11x + 2 = 0
  2. 2

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

    D=112−4⋅1⋅2=121−8=113D = 11^2 - 4\cdot1\cdot2 = 121 - 8 = 113
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−11±1132x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{113}}{2}
  5. 5

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

    x=−11+1132orx=−11−1132x = \frac{-11 + \sqrt{113}}{2}\quad\text{or}\quad x = \frac{-11 - \sqrt{113}}{2}
  6. 6

    As decimals: x ≈ -0.18492709 and x ≈ -10.815073.

  7. 7

    Check numerically: for x ≈ -0.18492709 the left side is 3.1086245e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 11x + 2 = 0\ \Rightarrow\ x = \frac{-11 + \sqrt{113}}{2},\ \frac{-11 - \sqrt{113}}{2} \]
\[ x^2 + 11x + 2 = 0 \]
\[ D = 11^2 - 4\cdot1\cdot2 = 121 - 8 = 113 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-11 \pm \sqrt{113}}{2} \]
\[ x = \frac{-11 + \sqrt{113}}{2}\quad\text{or}\quad x = \frac{-11 - \sqrt{113}}{2} \]

The parabola y = x² + 11x + 2

xy
-8.5-19.25
-7.5-24.25
-6.5-27.25
-5.5-28.25
-4.5-27.25
-3.5-24.25
-2.5-19.25

Questions

What are the solutions of x² + 11x + 2 = 0?

The solutions of x² + 11x + 2 = 0 are x = (-11 + √113)/2 or x = (-11 - √113)/2, that is x ≈ -0.18492709 and x ≈ -10.815073.

Does x² + 11x + 2 = 0 factor?

Not over the integers, because the discriminant 113 is not a perfect square. Use the quadratic formula.

What is the vertex of y = x² + 11x + 2?

(-11/2, -113/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -11, product = c/a = 2.

Similar equations

    Solve x² + 11x + 2 = 0: x = (-11 + √113)/2 or x = (-11 - √113)/2 (steps) | WordGPT