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Solve -2x² + 6x - 1 = 0

The solutions of -2x² + 6x - 1 = 0 are x = (3 + √7)/2 or x = (3 - √7)/2, that is x ≈ 0.17712434 and x ≈ 2.8228757.

x² +x += 0
−2x2+6x−1=0⇒x=3+72, 3−72-2x^2 + 6x - 1 = 0 \Rightarrow x = \frac{3 + \sqrt{7}}{2},\ \frac{3 - \sqrt{7}}{2}
Discriminant D = 28 · x ≈ 0.177124, 2.82288

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

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Graph of y = -2x² + 6x - 1-2-112345-25-20-15-10-55x = 0.17712434x = 2.8228757vertexy = -2x² + 6x - 1
Graph of y = -2x² + 6x - 1. The curve crosses the x-axis at x = 0.17712434 and x = 2.8228757; the vertex is at (1.5, 3.5).

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 = 6, c = -1.

    −2x2+6x−1=0-2x^2 + 6x - 1 = 0
  2. 2

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

    D=62−4⋅(−2)⋅(−1)=36−8=28D = 6^2 - 4\cdot(-2)\cdot(-1) = 36 - 8 = 28
  3. 3

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

  4. 4

    The quadratic formula works for every quadratic:

    x=−b±D2a=−6±28−4x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{28}}{-4}
  5. 5

    Simplify (pull the square out of the root) to get the solutions:

    x=3+72orx=3−72x = \frac{3 + \sqrt{7}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{7}}{2}
  6. 6

    As decimals: x ≈ 0.17712434 and x ≈ 2.8228757.

  7. 7

    Check numerically: for x ≈ 0.17712434 the left side is -3.3306691e-16 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ -2x^2 + 6x - 1 = 0\ \Rightarrow\ x = \frac{3 + \sqrt{7}}{2},\ \frac{3 - \sqrt{7}}{2} \]
\[ -2x^2 + 6x - 1 = 0 \]
\[ D = 6^2 - 4\cdot(-2)\cdot(-1) = 36 - 8 = 28 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{28}}{-4} \]
\[ x = \frac{3 + \sqrt{7}}{2}\quad\text{or}\quad x = \frac{3 - \sqrt{7}}{2} \]

The parabola y = -2x² + 6x - 1

xy
-1.5-14.5
-0.5-4.5
0.51.5
1.53.5
2.51.5
3.5-4.5
4.5-14.5

Questions

What are the solutions of -2x² + 6x - 1 = 0?

The solutions of -2x² + 6x - 1 = 0 are x = (3 + √7)/2 or x = (3 - √7)/2, that is x ≈ 0.17712434 and x ≈ 2.8228757.

Does -2x² + 6x - 1 = 0 factor?

Not over the integers, because the discriminant 28 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -2x² + 6x - 1?

(3/2, 7/2). The parabola opens downwards, so this is the maximum.

What are the sum and product of the roots?

Sum = −b/a = 3, product = c/a = 1/2.

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