WordGPT

Solve -2x² - x + 1 = 0

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

x² +x += 0
−2x2−x+1=0⇒x=−1, 12-2x^2 - x + 1 = 0 \Rightarrow x = -1,\ \frac{1}{2}
Discriminant D = 9 · x ≈ -1, 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 - x + 1-3-2-1123-20-15-10-5x = -1x = 0.5vertexy = -2x² - x + 1
Graph of y = -2x² - x + 1. The curve crosses the x-axis at x = -1 and x = 0.5; the vertex is at (-0.25, 1.125).

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 = -1, c = 1.

    −2x2−x+1=0-2x^2 - x + 1 = 0
  2. 2

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

    D=(−1)2−4⋅(−2)⋅1=1+8=9D = (-1)^2 - 4\cdot(-2)\cdot1 = 1 + 8 = 9
  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+1)(2x−1)=0-\left(x + 1\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+1=0⇒x=−12x−1=0⇒x=12x + 1 = 0 \Rightarrow x = -1\qquad 2x - 1 = 0 \Rightarrow x = \frac{1}{2}
  6. 6

    Method 2, The quadratic formula works for every quadratic:

    x=−b±D2a=1±9−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{9}}{-4}
  7. 7

    Simplify to get the solutions:

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

    Check by putting x = -1 back in: a·x² + b·x + c = -2·(1) + (-1)·(-1) + 1 = 0 ✓

LaTeX source of the formulas on this page
\[ -2x^2 - x + 1 = 0\ \Rightarrow\ x = -1,\ \frac{1}{2} \]
\[ -2x^2 - x + 1 = 0 \]
\[ D = (-1)^2 - 4\cdot(-2)\cdot1 = 1 + 8 = 9 \]
\[ -\left(x + 1\right)\left(2x - 1\right) = 0 \]
\[ x + 1 = 0 \Rightarrow x = -1\qquad 2x - 1 = 0 \Rightarrow x = \frac{1}{2} \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{1 \pm \sqrt{9}}{-4} \]
\[ x = -1\quad\text{or}\quad x = \frac{1}{2} \]

The parabola y = -2x² - x + 1

xy
-3.25-16.875
-2.25-6.875
-1.25-0.875
-0.251.125
0.75-0.875
1.75-6.875
2.75-16.875

Questions

What are the solutions of -2x² - x + 1 = 0?

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

Does -2x² - x + 1 = 0 factor?

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

What is the vertex of y = -2x² - x + 1?

(-1/4, 9/8). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = -1/2, product = c/a = -1/2.

Similar equations

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