Solve the following quadratic equation for all values of x in simplest form: 7 - x² = 6.

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Understand the Problem

The question is asking to solve a quadratic equation, specifically 7 - x² = 6, for all possible values of x. The solution needs to be presented in its simplest form.

Answer

The solutions are $x = 1$ and $x = -1$.
Answer for screen readers

The solutions to the equation $7 - x^2 = 6$ are $x = 1$ and $x = -1$.

Steps to Solve

  1. Rearrange the equation

First, we want to isolate the term with $x$. Start by moving 6 to the left side:

$$ 7 - 6 = x^2 $$

This simplifies to:

$$ 1 = x^2 $$

  1. Take the square root of both sides

To solve for $x$, take the square root of both sides. Remember that taking the square root produces both positive and negative solutions:

$$ x = \pm \sqrt{1} $$

Thus,

$$ x = 1 \quad \text{or} \quad x = -1 $$

  1. Present the solution

We now have the solutions for $x$. They are:

$$ x = 1 \quad \text{and} \quad x = -1 $$

The solutions to the equation $7 - x^2 = 6$ are $x = 1$ and $x = -1$.

More Information

Quadratic equations can often have two solutions because of the nature of square roots. In this case, both $1$ and $-1$ are valid because they satisfy the original equation.

Tips

  • Forgetting to include both the positive and negative roots when taking the square root.
  • Mistaking the operation when rearranging the equation (e.g., failing to correctly move terms).

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