Solve this quadratic equation by completing the square.x2 + 6x – 10 = 0A.B.C.D.

1. Break Down the Problem

To solve the quadratic equation $x^2 + 6x - 10 = 0$ by completing the square, we need to isolate the variable $x$.

2. Relevant Concepts

The method of completing the square involves transforming the quadratic equation into a perfect square trinomial. We use the following form:

$x^2 + bx + c = (x + d)^2 - e$

where $d$ and $e$ represent adjustments made to complete the square.

3. Analysis and Detail

Start with the equation:

$x^2 + 6x - 10 = 0$

1. Move the constant term to the other side:

   $x^2 + 6x = 10$

2. Complete the square:

   • Take half of the coefficient of $x$ (which is 6), and square it:

     $\left(\frac{6}{2}\right)^2 = 3^2 = 9$

   • Add 9 to both sides:

     $x^2 + 6x + 9 = 10 + 9$

     $(x + 3)^2 = 19$

4. Verify and Summarize

Now, we will take the square root of both sides:

$x + 3 = \pm \sqrt{19}$

Finally, isolate $x$:

$x = -3 \pm \sqrt{19}$

The two solutions can therefore be expressed as:

$x = -3 + \sqrt{19} \quad \text{and} \quad x = -3 - \sqrt{19}$

Final Answer

The solutions to the equation $x^2 + 6x - 10 = 0$ are:

$x = -3 + \sqrt{19} \quad \text{and} \quad x = -3 - \sqrt{19}$