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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 ≈ 4.4142136 and x ≈ 1.5857864.

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

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

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Graph of y = x² - 6x + 7123456-4-22468101214x = 4.4142136x = 1.5857864vertexy = x² - 6x + 7
Graph of y = x² - 6x + 7. The curve crosses the x-axis at x = 4.4142136 and x = 1.5857864; 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=0x^2 - 6x + 7 = 0
  2. 2

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

    D=(−6)2−4⋅1⋅7=36−28=8D = (-6)^2 - 4\cdot1\cdot7 = 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±82x = \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 ≈ 4.4142136 and x ≈ 1.5857864.

  7. 7

    Check numerically: for x ≈ 4.4142136 the left side is 0 (zero up to rounding) ✓

LaTeX source of the formulas on this page
\[ x^2 - 6x + 7 = 0\ \Rightarrow\ x = 3 + \sqrt{2},\ 3 - \sqrt{2} \]
\[ x^2 - 6x + 7 = 0 \]
\[ D = (-6)^2 - 4\cdot1\cdot7 = 36 - 28 = 8 \]
\[ x = \frac{-b \pm \sqrt{D}}{2a} = \frac{6 \pm \sqrt{8}}{2} \]
\[ x = 3 + \sqrt{2}\quad\text{or}\quad x = 3 - \sqrt{2} \]

The parabola y = x² - 6x + 7

xy
07
12
2-1
3-2
4-1
52
67

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 ≈ 4.4142136 and x ≈ 1.5857864.

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 upwards, so this is the minimum.

What are the sum and product of the roots?

Sum = −b/a = 6, product = c/a = 7.

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