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, 12-2x^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 + 6-8-6-4-22-40-30-20-101020x = -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=0-2x^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\cdot(-2)\cdot6 = 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±169−4x = \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
\[ -2x^2 - 11x + 6 = 0\ \Rightarrow\ x = -6,\ \frac{1}{2} \]
\[ -2x^2 - 11x + 6 = 0 \]
\[ D = (-11)^2 - 4\cdot(-2)\cdot6 = 121 + 48 = 169 \]
\[ -\left(x + 6\right)\left(2x - 1\right) = 0 \]
\[ x + 6 = 0 \Rightarrow x = -6\qquad 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{11 \pm \sqrt{169}}{-4} \]
\[ x = -6\quad\text{or}\quad x = \frac{1}{2} \]

The parabola y = -2x² - 11x + 6

xy
-5.753.125
-4.7513.125
-3.7519.125
-2.7521.125
-1.7519.125
-0.7513.125
0.253.125

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

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

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

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