Solving Quadratic Equations

Inspection, Square Roots, and Completing the Square

In this lesson:

• Solve equations in the form
• Use completing the square to reach that form

What You Will Be Able to Do

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

1. Solve by taking square roots
2. Complete the square to transform any quadratic into
3. Derive the quadratic formula; apply it to any quadratic
4. Factor quadratics and choose the best method
5. Write complex solutions in form

Recall: Square Roots and the ± Symbol

• : principal (positive) root —
• Squaring is two-to-one:
• Solving : undo both directions →

Why does have two solutions but have one?

Start with What You Already Know

Solve : what number squared gives 49?

• ✓ and ✓

Write both:

How many solutions does have when ?

Squaring Erases the Sign — So Write ±

Squaring maps both and to the same value:

Undoing a square means going back in both directions:

Solving the Form

Solve

or

Verify: ✓ and ✓

Two Special Cases to Know

Non-perfect square:

Double root: only

When Is Negative: No Real Solution

has no real solution.

for all real — it cannot equal .

In complex numbers: . Details in Lesson 2.

Quick Check: How Many Solutions?

For each equation, state the number of real solutions:

Solve equation 1 completely.

Rewriting to the Form We Can Solve

We can solve easily.

Most equations look like — not that form yet.

Goal: rewrite as .

That's exactly what completing the square does.

Why Completes the Square

, so means and the missing term is

Balance: Add to Both Sides

Adding to the left but not the right changes the equation:

Left side becomes . Right side updates too.

Whatever you do to one side, do to the other.

Completing the Square: Full Example

Solve:

Step 1: Move constant:

Step 2: Half of 8 is 4; . Add to both sides:

Step 3: Factor left side:

Step 4: Take :

Step 5: Solve:

Completing the Square: Negative Middle Term

Solve:

Move constant:

Half of is ; . Add to both sides:

What Went Wrong? Find and Fix

A student solved this way:

(added 9 to left only)

"No real solution" — Wrong!

What went wrong? What is the correct solution?

When Leading Coefficient Is Not One

Solve:

Step 0: Divide by 3:

Add to both sides:

Odd Values Produce Fractions

On : half of is ; squared is

The process is identical — fractions are normal.

Guided Practice: Completing the Square

Solve:

Step 1: Move constant: $x^2 - 6x = $ ___

Step 2: Half of ___ is ___. Squared: ___. Add to both sides.

Step 3: Factor left side as a perfect square: $(x - \square)^2 = $ ___

Step 4: Take : $x = $ ___ ___

Complete each step, then check with a partner.

Your Turn: Completing the Square

Solve by completing the square:

Show all five steps. Check by substituting one solution back.

Full Process from Blank Slate

Without scaffolding:

1. Solve

2. Solve

For each: choose the approach, execute, state the solution set.

What You Can Do Now

✓ Solve — write , solve for

✓ Complete the square — five steps to reach

Always write — both roots, not just the positive one

Add to both sides — not just the left side

Divide by first when leading coefficient isn't 1

Coming Up: The Quadratic Formula

Lesson 2: Complete the square on — with letters instead of numbers.

The result is a formula that solves every quadratic automatically.

What you built today is exactly what Lesson 2 uses.