WordGPT

Solve 2x² - 11x - 6 = 0

The solutions of 2x² - 11x - 6 = 0 are x = 6 or x = -1/2.

x² +x += 0
2x2−11x−6=0⇒x=6, −122x^2 - 11x - 6 = 0 \Rightarrow x = 6,\ -\frac{1}{2}
Discriminant D = 169 · x ≈ 6, -0.5

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 - 11x - 6-22468-20-101020304050x = 6x = -0.5vertexy = 2x² - 11x - 6
Graph of y = 2x² - 11x - 6. The curve crosses the x-axis at x = 6 and x = -0.5; the vertex is at (2.75, -21.125).

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 = -11, c = -6.

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

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

    D=(−11)2−4⋅2⋅(−6)=121+48=169D = (-11)^2 - 4\cdot2\cdot(-6) = 121 + 48 = 169
  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:

    (x−6)(2x+1)=0\left(x - 6\right)\left(2x + 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:

    x−6=0⇒x=62x+1=0⇒x=−12x - 6 = 0 \Rightarrow x = 6\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=11±1694x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{4}
  7. 7

    Simplify to get the solutions:

    x=6orx=−12x = 6\quad\text{or}\quad x = -\frac{1}{2}
  8. 8

    Check by putting x = 6 back in: a·x² + b·x + c = 2·(36) + (-11)·(6) + (-6) = 0 ✓

LaTeX source of the formulas on this page
\[ 2x^2 - 11x - 6 = 0\ \Rightarrow\ x = 6,\ -\frac{1}{2} \]
\[ 2x^2 - 11x - 6 = 0 \]
\[ D = (-11)^2 - 4\cdot2\cdot(-6) = 121 + 48 = 169 \]
\[ \left(x - 6\right)\left(2x + 1\right) = 0 \]
\[ x - 6 = 0 \Rightarrow x = 6\qquad 2x + 1 = 0 \Rightarrow x = -\frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{4} \]
\[ x = 6\quad\text{or}\quad x = -\frac{1}{2} \]

The parabola y = 2x² - 11x - 6

xy
-0.25-3.125
0.75-13.125
1.75-19.125
2.75-21.125
3.75-19.125
4.75-13.125
5.75-3.125

Questions

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

The solutions of 2x² - 11x - 6 = 0 are x = 6 or x = -1/2.

Does 2x² - 11x - 6 = 0 factor?

Yes: (x-6)(2x+1) = 0.

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

(11/4, -169/8). The parabola opens upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 11/2, product = c/a = -3.

Similar equations

    Solve 2x² - 11x - 6 = 0: x = 6 or x = -1/2 (steps) | WordGPT