Definition of the square root property in solving quadratic equations

📚 Definition of the Square Root Property

The square root property is a technique used to solve quadratic equations of the form $x^2 = k$, where $x$ is a variable and $k$ is a constant. It states that if $x^2 = k$, then $x = \pm \sqrt{k}$. This means $x$ can be either the positive or negative square root of $k$.

📜 History and Background

The concept of solving equations by taking roots dates back to ancient mathematics. Early mathematicians in Babylon and Egypt dealt with problems that are now recognized as quadratic equations. However, the explicit formulation of the square root property as a general technique evolved over centuries, alongside the development of algebraic notation.

💡 Key Principles

• 🔢 Isolate the Squared Term: The first step is to isolate the squared term (e.g., $x^2$) on one side of the equation.
• ➕ Apply the Square Root: Take the square root of both sides of the equation. Remember to include both the positive and negative roots.
• ✅ Simplify: Simplify the square roots if possible.
• 🎯 Solve for the Variable: Solve for the variable to find the possible solutions.

➗ Real-world Examples

Example 1: Solve $x^2 = 9$

1. Take the square root of both sides: $\sqrt{x^2} = \pm \sqrt{9}$
2. Simplify: $x = \pm 3$
3. Solutions: $x = 3$ or $x = -3$

Example 2: Solve $(x - 2)^2 = 16$

1. Take the square root of both sides: $\sqrt{(x - 2)^2} = \pm \sqrt{16}$
2. Simplify: $x - 2 = \pm 4$
3. Solve for $x$: $x = 2 \pm 4$
4. Solutions: $x = 6$ or $x = -2$

Example 3: Solve $4x^2 - 25 = 0$

1. Isolate the squared term: $4x^2 = 25$
2. Divide by 4: $x^2 = \frac{25}{4}$
3. Take the square root of both sides: $\sqrt{x^2} = \pm \sqrt{\frac{25}{4}}$
4. Simplify: $x = \pm \frac{5}{2}$
5. Solutions: $x = \frac{5}{2}$ or $x = -\frac{5}{2}$

📝 Practice Quiz

Solve the following equations using the square root property:

1. $x^2 = 49$
2. $(x + 1)^2 = 25$
3. $9x^2 = 16$
4. $(2x - 3)^2 = 1$
5. $x^2 - 81 = 0$
6. $(x - 5)^2 = 36$
7. $16x^2 - 9 = 0$

🔑 Conclusion

The square root property is a powerful tool for solving quadratic equations, especially when the equation can be easily written in the form $x^2 = k$. Understanding this property provides a straightforward method for finding solutions and builds a strong foundation for more advanced algebraic techniques.