Solve equations using square roots
To undo squaring a variable (x²), you use its inverse operation: taking the square root. This is a key tool for solving equations involving squares, like those found in the Pythagorean theorem or area problems.
Do this: Read the concept below, then try the quiz or activity.
Concept
When an equation involves a variable squared (like x²), you can solve it by isolating the squared term and then taking the square root of both sides.
The Fundamental Rule If x² = a, then x = ±√a. This means x can be the positive square root of 'a' OR the negative square root of 'a'.
Why are there two solutions? Because both a positive number and its negative opposite will result in a positive number when squared. * 3² = 9 * (-3)² = 9 So, if x² = 9, x could be 3 or -3.
Steps to Solve:
1. Isolate the squared term (the variable part) on one side of the equation. 2. Take the square root of both sides of the equation. 3. Remember to include both the positive and negative solutions (±). 4. Solve for the variable if necessary.
Example 1: A simple case
* x² = 49
1. The squared term is already isolated.
2. Take the square root of both sides: √x² = √49
3. x = ±7
* The solutions are x = 7 and x = -7.Example 2: A two-step equation
* 3x² - 12 = 36
1. Isolate x²:
* Add 12 to both sides: 3x² = 48
* Divide by 3: x² = 16
2. Take the square root:
* √x² = √16
3. Remember both solutions:
* x = ±4
* The solutions are x = 4 and x = -4.What if the number is not a perfect square? You can leave the answer in square root form (as a radical). * x² = 10 * x = ±√10
No Solution? You cannot take the square root of a negative number in basic algebra. * x² = -25 * This equation has no real solution.
Key Idea: Taking the square root is the inverse operation of squaring something. They cancel each other out, just like addition and subtraction do.
Try it
Practice: Solve equations using square roots.