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.
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.