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Solve -2x² + x + 1 = 0

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

x² +x += 0
−2x2+x+1=0⇒x=−12, 1-2x^2 + x + 1 = 0 \Rightarrow x = -\frac{1}{2},\ 1
Discriminant D = 9 · x ≈ -0.5, 1

Change a, b or c (whole numbers, a not 0) and the exact solutions follow.

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Graph of y = -2x² + x + 1-3-2-1123-20-15-10-5x = -0.5x = 1vertexy = -2x² + x + 1
Graph of y = -2x² + x + 1. The curve crosses the x-axis at x = -0.5 and x = 1; 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=12−4⋅(−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:

    −(2x+1)(x−1)=0-\left(2x + 1\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:

    2x+1=0⇒x=−12x−1=0⇒x=12x + 1 = 0 \Rightarrow x = -\frac{1}{2}\qquad x - 1 = 0 \Rightarrow x = 1
  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=−12orx=1x = -\frac{1}{2}\quad\text{or}\quad x = 1
  8. 8

    Check by putting x = -1/2 back in: a·x² + b·x + c = -2·(1/4) + 1·(-1/2) + 1 = 0 ✓

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

The parabola y = -2x² + x + 1

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

Questions

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

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

Does -2x² + x + 1 = 0 factor?

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

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