WordGPT

Solve 2x² - 6x - 11 = 0

The solutions of 2x² - 6x - 11 = 0 are x = (3 + √31)/2 or x = (3 - √31)/2, that is x ≈ 4.2838822 and x ≈ -1.2838822.

x² +x += 0
2x2−6x−11=0⇒x=3+312, 3−3122x^2 - 6x - 11 = 0 \Rightarrow x = \frac{3 + \sqrt{31}}{2},\ \frac{3 - \sqrt{31}}{2}
Discriminant D = 124 · x ≈ 4.28388, -1.28388

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 - 6x - 11-2246-20-1010203040x = 4.2838822x = -1.2838822vertexy = 2x² - 6x - 11
Graph of y = 2x² - 6x - 11. The curve crosses the x-axis at x = 4.2838822 and x = -1.2838822; the vertex is at (1.5, -15.5).

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 = -6, c = -11.

    2x2−6x−11=02x^2 - 6x - 11 = 0
  2. 2

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

    D=(−6)2−4⋅2⋅(−11)=36+88=124D = (-6)^2 - 4\cdot2\cdot(-11) = 36 + 88 = 124
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=6±1244x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{124}}{4}
  5. 5

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

    x=3+312orx=3−312x = \frac{3 + \sqrt{31}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{31}}{2}
  6. 6

    As decimals: x ≈ 4.2838822 and x ≈ -1.2838822.

  7. 7

    Check numerically: for x ≈ 4.2838822 the left side is 3.5527137e-15 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 6x - 11 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{31}}{2},\ \frac{3 - \sqrt{31}}{2} \]
\[ 2x^2 - 6x - 11 = 0 \]
\[ D = (-6)^2 - 4\cdot2\cdot(-11) = 36 + 88 = 124 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{124}}{4} \]
\[ x = \frac{3 + \sqrt{31}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{31}}{2} \]

The parabola y = 2x² - 6x - 11

xy
-1.52.5
-0.5-7.5
0.5-13.5
1.5-15.5
2.5-13.5
3.5-7.5
4.52.5

Questions

What are the solutions of 2x² - 6x - 11 = 0?

The solutions of 2x² - 6x - 11 = 0 are x = (3 + √31)/2 or x = (3 - √31)/2, that is x ≈ 4.2838822 and x ≈ -1.2838822.

Does 2x² - 6x - 11 = 0 factor?

Not over the integers, because the discriminant 124 is not a perfect square. Use the quadratic formula.

What is the vertex of y = 2x² - 6x - 11?

(3/2, -31/2). 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 = -11/2.

Similar equations

    Solve 2x² - 6x - 11 = 0: x = (3 + √31)/2 or x = (3 - √31)/2 (steps) | WordGPT