Lesson 1.11

Solving by Square Roots

When a quadratic equation lacks a linear "x" term, we don't need to factor. We can simply use the inverse of squaring: the square root.

Introduction

Just as division undoes multiplication, square roots undo exponents. This method is incredibly fast but only works for specific types of equations: those where the variable is contained entirely within a squared term.

Past Knowledge

You know and .

Today's Goal

We solve by taking the square root of both sides, remembering BOTH positive and negative answers.

Future Success

This is the foundation for "Completing the Square" (Chapter 4), which leads to the Quadratic Formula.

Key Concepts

The Inverse Property

If , then:

You must include the symbol. Every positive number has two square roots.

The Strategy: Isolate

  1. Isolate: Get the squared part by itself.
  2. Root: Take of both sides. Don't forget .
  3. Simplify: Reduce radical if needed.

Worked Examples

Example 1: Basic Solving

Basic

Solve .

1

Isolate $x^2$

Divide both sides by 3.

2

Square Root

Solutions:

Example 2: Irrational Solutions

Intermediate

Solve .

1

Isolate $x^2$

Add 10, then divide by 2.

2

Square Root

Result: (approx)

Example 3: Grouped Squared Term

Advanced

Solve .

1

Root First!

Do not FOIL! The squared term is already isolated on the left.

2

Split and Solve

Two cases: and .

Result:

Common Pitfalls

Forgetting Plus-Minus

If , is only half the answer. is also 9. You typically need 2 solutions for a quadratic.

Negative Radicals

has NO REAL SOLUTION (yet). You cannot square a real number and get a negative. (See Chapter 3 for Complex Numbers).

Real-Life Applications

Falling Objects

The time for an object to fall a distance is approx (in feet).

To find time from distance, you must use square roots. If a rock falls 64 ft, seconds. (We ignore because time cannot be negative).

Practice Quiz

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