4. Consider the problem . Solve by Factoring, Completing the Square, the Quadratic
   Formula. Show all work.

To solve the quadratic equation by factoring, we first look for two numbers that multiply to and add to . The numbers and work since and .

We can rewrite as :

Next, we group and factor:

Factoring out common terms:

Combining gives:

Setting each factor to zero:

Now, we’ll complete the square. Start with the given equation:

Divide everything by 2:

Now, isolate the constant term:

To complete the square, take half of the coefficient of , square it, and add to both sides:

Adding to both sides:

Taking the square root of both sides:

Solving for :

Lastly, let’s apply the quadratic formula:

Substitute :

Now plug this into the formula:

This gives:

So, the solutions are and .

Want to know how this work relates to math history? The quadratic formula, used in this problem, dates back to the ancient Babylonians, who solved similar equations using geometric methods. They didn’t have algebra as we know it, but they were able to manipulate numbers to find the roots of equations—pretty cool right?

Now, for some real-world application: Quadratic equations pop up everywhere! From determining the height of a projectile in physics to optimizing area in business models—knowing how to solve these can help you tackle tons of practical problems. Imagine optimizing a garden layout or maximizing profit—thanks, quadratic!