Solving Quadratics: Formula and Method Selection

Lesson 2 of 2

In this lesson:

• Derive the quadratic formula from completing the square
• Apply and choose among solving methods
• Recognize and write complex solutions

What You Will Be Able to Do

By the end of both lessons, you should be able to:

1. Solve using square roots
2. Complete the square on any quadratic
3. Derive and apply the quadratic formula
4. Choose the best method per equation
5. Write complex solutions in

Bridge: Completing the Square with Letters

Last lesson:

Today: replace 6 with , replace 2 with , replace 1 with

What if we ran the same five steps with letters instead of numbers?

Derivation Step 1: Divide by

Start:

Divide everything by :

Move the constant:

Derivation Step 2: Complete the Square

Half of is ; squared is . Add to both sides:

Derivation Step 3: Arrive at the Formula

From , take :

The quadratic formula — completing the square, done once for all.

Applying the Quadratic Formula Correctly

1. Label , , before substituting
2. Compute first
3. Simplify both the and cases

Works for every ,

Apply the Formula: Integer Solutions

Solve:   ()

Verify: ✓

Apply the Formula: Irrational Solutions

Solve:

Does this factor over the integers? No — use the formula.

Check-In: Identify , , and the Discriminant

For :

1. Write
2. Compute
3. How many real solutions does this equation have?

Solve completely.

Using the Discriminant to Classify Solutions

: two real solutions   : one (double)   : complex

Why Factor When the Formula Works?

The formula always works — so why use any other method?

The formula is a reliable backup, not the first choice.

Choose a Method and Solve

Choose the method and solve:

For each: name the method, execute, state the solution set.

When the Discriminant Is Negative

has no real solution — but solutions do exist.

Define: , so

Complex solutions use ; they are valid, just not real.

Complex Solutions in Form

Solve:   ()

Solutions: and

Complex Solutions Come in Conjugate Pairs

If is a solution, then is also a solution.

Verify in :

✓

Guided Practice: Solving with Complex Results

Solve:

Step 1:

Step 2: $b^2 - 4ac = $ ____

Step 3:

Step 4: $x = $ ____ in form

Discriminant Detective: Predict Then Solve

Compute , predict the type, solve only if real.

Discriminant first — then solve.

Full Process from Blank Slate

No hints. Choose your method and solve completely.

For each: state solution type, show work, write solution set.

What You Can Do Now

✓ Derive the quadratic formula from completing the square

✓ Label , , ; compute discriminant; apply the formula

✓ Choose the fastest method for each equation

Label , , before substituting — sign errors start here

Negative discriminant → complex solutions, not "no solution"

Coming Up: Solving Linear-Quadratic Systems

Next: HSA.REI.C.7 — what happens when a linear and a quadratic equation are both true at once?

The intersection points of a line and a parabola satisfy both equations.

Solving those systems uses the quadratic-solving skills you have now.