complete the square to identify the two values that solve the following quadratic equation x squared - 16x - 13 equals 14

Answer 1

To solve the quadratic equation $x^2-16x-13=14$ by completing the square, first, we need to rearrange the equation into a standard form:

1. Subtract 14 from both sides:

   $$x^2-16x-13-14=0$$

   This simplifies to:

   $$x^2-16x-27=0$$

2. Now, let's complete the square. We want to express the quadratic in the form $(x-p{)}^2=q$.

3. First, take the coefficient of $x$, which is $-16$, halve it, and then square it:

   $${\left(\frac{-16}{2}\right)}^2=(-8{)}^2=64$$

4. Next, add and subtract $64$ to the left-hand side of the equation:

   $$x^2-16x+64-64-27=0$$

   This can be rearranged to:

   $$(x-8{)}^2-91=0$$

5. Now isolate the squared term:

   $$(x-8{)}^2=91$$

6. Take the square root of both sides:

   $$x-8=\pm \sqrt{91}$$

7. Finally, solve for $x$:

   $$x=8\pm \sqrt{91}$$

Thus, the two values that solve the equation are:

$$x=8+\sqrt{91}\text{and}x=8-\sqrt{91}$$