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Solve -x² + 6x - 7 = 0

The solutions of -x² + 6x - 7 = 0 are x = 3 + √2 or x = 3 - √2, that is x ≈ 1.5857864 and x ≈ 4.4142136.

x² +x += 0
−x2+6x−7=0⇒x=3+2, 3−2-x^2 + 6x - 7 = 0 \Rightarrow x = 3 + \sqrt{2},\ 3 - \sqrt{2}
Discriminant D = 8 · x ≈ 1.58579, 4.41421

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

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Graph of y = -x² + 6x - 7123456-14-12-10-8-6-4-224x = 1.5857864x = 4.4142136vertexy = -x² + 6x - 7
Graph of y = -x² + 6x - 7. The curve crosses the x-axis at x = 1.5857864 and x = 4.4142136; the vertex is at (3, 2).

How to work it out

  1. 1

    Write it in standard form a x² + b x + c = 0 and read off the coefficients: a = -1, b = 6, c = -7.

    −x2+6x−7=0-x^2 + 6x - 7 = 0
  2. 2

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

    D=62−4⋅(−1)⋅(−7)=36−28=8D = 6^2 - 4\cdot(-1)\cdot(-7) = 36 - 28 = 8
  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±8−2x = \frac{-b \pm \sqrt{D}}{2a} = \frac{-6 \pm \sqrt{8}}{-2}
  5. 5

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

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

    As decimals: x ≈ 1.5857864 and x ≈ 4.4142136.

  7. 7

    Check numerically: for x ≈ 1.5857864 the left side is -8.8817842e-16 (zero up to rounding) ✓

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

The parabola y = -x² + 6x - 7

xy
0-7
1-2
21
32
41
5-2
6-7

Questions

What are the solutions of -x² + 6x - 7 = 0?

The solutions of -x² + 6x - 7 = 0 are x = 3 + √2 or x = 3 - √2, that is x ≈ 1.5857864 and x ≈ 4.4142136.

Does -x² + 6x - 7 = 0 factor?

Not over the integers, because the discriminant 8 is not a perfect square. Use the quadratic formula.

What is the vertex of y = -x² + 6x - 7?

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

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = 7.

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    Solve -x² + 6x - 7 = 0: x = 3 + √2 or x = 3 - √2 (steps) | WordGPT