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Radicals Introduction

An equation that contains roots involving variables, such as √(x-5) =2 is called a radical equation. The main goal in solving them is to get rid of roots and to convert the radical equation to some simple, known equation.

Product rule for radicals:

n√a . n√b = n√ab

Quotient rule for radicals:

n√a / n√b = n√a/b

Adding and Subtracting Radical Expressions:

We can add or subtract radicals using the distributive property.

Example: 5√3 +2√3

= (5+2)√3

= 7√3

Multiplying Radical Expressions:

Using special product rule with radicals:

(a-b) .(a+b) = a² +b²

Solving an equation with radicals:

• Isolate the radical (or at least one of the radicals if there are more than one).
• Square both sides
• Combine like terms
• Repeat steps 1-3 until no radicals are remaining
• Solve the equation
• Check all solutions with the original equation (some may not work)

Example

x= √(3x+7) -1

Adding 1on both side x+1 = √(3x+7)

Squaring both side x²+2x+1=3x+7

Subtracting 3x+7 from both side:

x² + x-6=0

(x-3)(x+2) =0

So x = -2 and x = 3, but only x =3 makes the original equation equal.

Dividing by a complex number: