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Complete the square: x^2 + 12x - 14 = 0

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5 answers

To complete the square for the equation \( x^2 + 12x - 14 = 0 \), follow these steps:

  1. Start with the equation: \[ x^2 + 12x - 14 = 0 \]

  2. Move the constant term to the other side of the equation: \[ x^2 + 12x = 14 \]

  3. To complete the square, take half of the coefficient of \( x \) (which is 12), square it, and add it to both sides. Half of 12 is 6, and squaring it gives: \[ 6^2 = 36 \]

  4. Add 36 to both sides: \[ x^2 + 12x + 36 = 14 + 36 \]

  5. Simplify the right side: \[ x^2 + 12x + 36 = 50 \]

  6. Now, the left side can be written as a square: \[ (x + 6)^2 = 50 \]

  7. Finally, take the square root of both sides: \[ x + 6 = \pm \sqrt{50} \]

  8. Simplifying \(\sqrt{50}\): \[ \sqrt{50} = 5\sqrt{2} \]

  9. Thus, we have: \[ x + 6 = 5\sqrt{2} \quad \text{or} \quad x + 6 = -5\sqrt{2} \]

  10. Solving for \( x \): \[ x = -6 + 5\sqrt{2} \quad \text{or} \quad x = -6 - 5\sqrt{2} \]

So the solutions to the equation \( x^2 + 12x - 14 = 0 \) are: \[ x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2} \]

can \(-6\pm\sqrt{50}\) be an answer?

Let's clarify the results from the previous solution.

We started with the equation:

\[ x^2 + 12x - 14 = 0 \]

After completing the square, we derived:

\[ (x + 6)^2 = 50 \]

Taking the square root of both sides gives:

\[ x + 6 = \pm \sqrt{50} \]

This means:

\[ x + 6 = \sqrt{50} \quad \text{or} \quad x + 6 = -\sqrt{50} \]

Now, solving for \(x\):

  1. From \(x + 6 = \sqrt{50}\): \[ x = -6 + \sqrt{50} \]

  2. From \(x + 6 = -\sqrt{50}\): \[ x = -6 - \sqrt{50} \]

Now, simplifying \(\sqrt{50}\):

\[ \sqrt{50} = 5\sqrt{2} \]

This means the solutions can also be expressed as:

  1. \(x = -6 + 5\sqrt{2}\)
  2. \(x = -6 - 5\sqrt{2}\)

So the values \(-6 \pm \sqrt{50}\) can be used as valid forms. However, for clarity and standard presentation, typically \(5\sqrt{2}\) would be preferred when simplifying \(\sqrt{50}\).

Thus, to answer your question directly, \(-6 \pm \sqrt{50}\) can indeed be an answer, but it is better expressed as \(-6 \pm 5\sqrt{2}\) when fully simplified.

what other answer choice can be considered when isolating the variable?

When isolating the variable in the equation \(x^2 + 12x - 14 = 0\), and after completing the square, we reached the expression:

\[ (x + 6)^2 = 50 \]

Taking the square root of both sides yields:

\[ x + 6 = \pm \sqrt{50} \]

Now, isolating \(x\) gives us:

\[ x = -6 + \sqrt{50} \quad \text{or} \quad x = -6 - \sqrt{50} \]

Here are some equivalent representations that could also be considered when isolating the variable:

  1. Expressing within different radical forms:

    • You can keep the expression as \(-6 + \sqrt{50}\) and \(-6 - \sqrt{50}\).
    • Alternatively, expressing each square root as simplified would yield: \[ x = -6 + 5\sqrt{2} \quad \text{and} \quad x = -6 - 5\sqrt{2} \]

  2. Numerical approximations: If further numerical clarity is desired, you could also calculate approximate decimal values. Since \(\sqrt{50} \approx 7.07\): \[ x \approx -6 + 7.07 \approx 1.07 \quad \text{and} \quad x \approx -6 - 7.07 \approx -13.07 \]

Aside from these options, the general forms remain valid:

  • \(-6 + \sqrt{50}\)
  • \(-6 - \sqrt{50}\)
  • \(-6 + 5\sqrt{2}\)
  • \(-6 - 5\sqrt{2}\)

Any of these forms can be used as answers when isolating the variable in this quadratic equation. The best choice depends on context—whether you need exact forms, simplified forms, or numerical approximations.