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 up to . These numbers are and . We can rewrite the equation as:

Next, we factor by grouping:

This simplifies to:

Factoring out the common term :

Setting each factor equal to zero gives:

Now, completing the square:

Start with . Divide through by 2:

Rearranging gives:

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

So we have:

This simplifies to:

Taking the square root:

Solving for , we find:

Finally, using the quadratic formula :

Here :

This yields:

All methods give us the solutions and .

Did you know that the quadratic formula was developed over hundreds of years? The ancient Greeks had their version, using geometric methods, long before the formula took its algebraic shape! Further enhancements in the Middle Ages led to deeper insights that formed the basis for what we use today.

In practice, solving quadratics using different methods can save you a lot of time and effort! If the equation is factorable, it’s often quicker to factor rather than using the quadratic formula. However, remember that completing the square always works, and is particularly helpful in calculus when you’re dealing with vertex forms!