Mathematics - 10th grade · Radical expressions
3
Radical expressions
Interactive lesson - ~8 min - ★★☆
Lesson progress
🧵In this lesson we follow one expression, √(x - 2), from start to finish: how it behaves, when it is defined, what it means geometrically, and how to solve an equation with it.
1What is a radical expression?
Some expressions contain a root with a variable underneath. They are called radical expressions - for example √(x - 2) or ∛(2a - 1). Unlike √9 = 3, their value depends on what we substitute for the variable.
Definition
A radical expression is an algebraic expression in which a variable sits under a root sign.
2How does √x behave?
Before working with √(x - 2), let's look at the basis - √x. Move the slider and watch the root grow. Try going below zero.
Notice
When x < 0, the point disappears - over the real numbers √x is undefined for negative numbers. We will need this condition right away.
3When is √(x - 2) defined?
For √(x - 2) the radicand must be ≥ 0, so x - 2 ≥ 0, meaning x ≥ 2. Drag the point and find the green zone where the expression is defined.
4Geometric meaning of the root
The root √a is the side of a square with area a. Change the area and see how the side is exactly its square root - that's why taking a root “undoes” the square.
5Solve with me: √(x - 2) = 3
Now we solve an equation with the same expression. Tap each step to reveal it - but try it yourself first.
1Check the domain condition›
x - 2 ≥ 0 → x ≥ 2. We look for a solution only in this region.
2Square both sides›
(√(x - 2))² = 3² → x - 2 = 9
3Solve the linear equation›
x = 9 + 2 → x = 11
4Check the solution›
x = 11 ≥ 2 ✓ and √(11 - 2) = √9 = 3 ✓ - correct!
6Quick check
For which values of x is the expression √(x - 5) defined?
Afor every real x
Bx ≤ 5
Cx ≥ 5
Donly x = 5
👁What the teacher sees
In a real lesson the teacher sees how long each step took, which were revealed, and where the student erred - not just the final answer.
✓
Lesson completed!
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