WordGPT

Solve 5x² - 6x - 11 = 0

The solutions of 5x² - 6x - 11 = 0 are x = 11/5 or x = -1.

x² +x += 0
5x2−6x−11=0⇒x=115, −15x^2 - 6x - 11 = 0 \Rightarrow x = \frac{11}{5},\ -1
Discriminant D = 256 · x ≈ 2.2, -1

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 = 5x² - 6x - 11-3-2-11234-20204060x = 2.2x = -1vertexy = 5x² - 6x - 11
Graph of y = 5x² - 6x - 11. The curve crosses the x-axis at x = 2.2 and x = -1; the vertex is at (0.6, -12.8).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = 5, b = -6, c = -11.

    5x2−6x−11=05x^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⋅5⋅(−11)=36+220=256D = (-6)^2 - 4\cdot5\cdot(-11) = 36 + 220 = 256
  3. 3

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

  4. 4

    Method 1, factoring. The roots are rational, so the left side factors:

    (5x−11)(x+1)=0\left(5x - 11\right)\left(x + 1\right) = 0
  5. 5

    A product is zero when one of its factors is zero, so set each factor equal to 0 and solve:

    5x−11=0⇒x=115x+1=0⇒x=−15x - 11 = 0 \Rightarrow x = \frac{11}{5}\qquad x + 1 = 0 \Rightarrow x = -1
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=6±25610x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{256}}{10}
  7. 7

    Simplify to get the solutions:

    x=115orx=−1x = \frac{11}{5}\quad\text{or}\quad x = -1
  8. 8

    Check by putting x = 11/5 back in: a·x² + b·x + c = 5·(121/25) + (-6)·(11/5) + (-11) = 0 ✓

LaTeX source of the formulas on this page
\[ 5x^2 - 6x - 11 = 0\ \Rightarrow\ x = \frac{11}{5},\ -1 \]
\[ 5x^2 - 6x - 11 = 0 \]
\[ D = (-6)^2 - 4\cdot5\cdot(-11) = 36 + 220 = 256 \]
\[ \left(5x - 11\right)\left(x + 1\right) = 0 \]
\[ 5x - 11 = 0 \Rightarrow x = \frac{11}{5}\qquad x + 1 = 0 \Rightarrow x = -1 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{256}}{10} \]
\[ x = \frac{11}{5}\quad\text{or}\quad x = -1 \]

The parabola y = 5x² - 6x - 11

xy
-2.432.2
-1.47.2
-0.4-7.8
0.6-12.8
1.6-7.8
2.67.2
3.632.2

Questions

What are the solutions of 5x² - 6x - 11 = 0?

The solutions of 5x² - 6x - 11 = 0 are x = 11/5 or x = -1.

Does 5x² - 6x - 11 = 0 factor?

Yes: (5x-11)(x+1) = 0.

What is the vertex of y = 5x² - 6x - 11?

(3/5, -64/5). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6/5, product = c/a = -11/5.

Similar equations

    Solve 5x² - 6x - 11 = 0: x = 11/5 or x = -1 (steps) | WordGPT