WordGPT

Solve x² + 3x - 11 = 0

The solutions of x² + 3x - 11 = 0 are x = (-3 + √53)/2 or x = (-3 - √53)/2, that is x ≈ 2.1400549 and x ≈ -5.1400549.

x² +x += 0
x2+3x−11=0⇒x=−3+532, −3−532x^2 + 3x - 11 = 0 \Rightarrow x = \frac{-3 + \sqrt{53}}{2},\ \frac{-3 - \sqrt{53}}{2}
Discriminant D = 53 · x ≈ 2.14005, -5.14005

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 - 11-6-4-224-101020x = 2.1400549x = -5.1400549vertexy = x² + 3x - 11
Graph of y = x² + 3x - 11. The curve crosses the x-axis at x = 2.1400549 and x = -5.1400549; the vertex is at (-1.5, -13.25).

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 = 3, c = -11.

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

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

    D=32−4⋅1⋅(−11)=9+44=53D = 3^2 - 4\cdot1\cdot(-11) = 9 + 44 = 53
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−3±532x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{53}}{2}
  5. 5

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

    x=−3+532orx=−3−532x = \frac{-3 + \sqrt{53}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{53}}{2}
  6. 6

    As decimals: x ≈ 2.1400549 and x ≈ -5.1400549.

  7. 7

    Check numerically: for x ≈ 2.1400549 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 + 3x - 11 = 0\ \Rightarrow\ x = \frac{-3 + \sqrt{53}}{2},\ \frac{-3 - \sqrt{53}}{2} \]
\[ x^2 + 3x - 11 = 0 \]
\[ D = 3^2 - 4\cdot1\cdot(-11) = 9 + 44 = 53 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-3 \pm \sqrt{53}}{2} \]
\[ x = \frac{-3 + \sqrt{53}}{2}\quad\text{or}\quad x = \frac{-3 - \sqrt{53}}{2} \]

The parabola y = x² + 3x - 11

xy
-4.5-4.25
-3.5-9.25
-2.5-12.25
-1.5-13.25
-0.5-12.25
0.5-9.25
1.5-4.25

Questions

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

The solutions of x² + 3x - 11 = 0 are x = (-3 + √53)/2 or x = (-3 - √53)/2, that is x ≈ 2.1400549 and x ≈ -5.1400549.

Does x² + 3x - 11 = 0 factor?

Not over the integers, because the discriminant 53 is not a perfect square. Use the quadratic formula.

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

(-3/2, -53/4). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = -3, product = c/a = -11.

Similar equations

    Solve x² + 3x - 11 = 0: x = (-3 + √53)/2 or x = (-3 - √53)/2 (steps) | WordGPT