Problem 27 - Algebra A Combined Function

Question

Solve. $$ x^{2}=16 $$

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = -4\).

Step 1: Understand the Equation

First, look at the given equation: \(x^2 = 16\). This is a quadratic equation where you need to find the values of \(x\) that make this equation true.

Step 2: Take the Square Root

Since the equation is \(x^2 = 16\), solve for \(x\) by taking the square root of both sides. This gives two possible solutions for \(x\), since both \(x\) and \(-x\) can be squared to equal \(16\). Therefore, \(x = \sqrt{16}\) or \(x = -\sqrt{16}\).

Step 3: Simplify the Square Roots

Calculate the square roots from the previous step: \(\sqrt{16} = 4\) and \(-\sqrt{16} = -4\). Therefore, the two solutions for \(x\) are \(x = 4\) and \(x = -4\).

Step 4: Verify the Solutions

Verify the solutions by substituting them back into the original equation. For \(x = 4\), \(4^2 = 16\), which holds true. For \(x = -4\), \((-4)^2 = 16\), which also holds true. Both solutions are correct.

Key Concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions of the form \( ax^2 + bx + c = 0 \), but sometimes they can appear simpler, like the one we're discussing: \( x^2 = 16 \). Solving equations involves finding the unknown number or numbers, represented by variables like \( x \), that satisfy the equation.
In our example, the task is to find values of \( x \) that make \( x^2 = 16 \) a true statement. This specific equation is a type of quadratic equation because it involves \( x \) multiplied by itself (squared).
Here's a streamlined process for solving basic quadratic equations of this form:

• Recognize the equation as a simple quadratic, where one side is a perfect square and the other side is a constant.
• Understand that there will typically be two solutions for \( x \), corresponding to the positive and negative square roots of the constant.
• Pursue solving by removing the square through mathematical operations – specifically, taking the square root of both sides.

Always remember, solving equations is like unraveling a puzzle, seeking the values of \( x \) that solve the problem.

Understanding Square Roots

Square roots are the numbers that, when multiplied by themselves, give the original number. In the equation \( x^2 = 16 \), the square root is the number \( x \) must be to satisfy this equation.
Since squaring either positive or negative values results in a positive number (like \( 4^2 = 16 \) and \( (-4)^2 = 16 \)), the square root of 16 has both a positive and a negative value.
Taking the square root of both sides of the equation \( x^2 = 16 \) gives us:

• \( x = \sqrt{16} \) which equals 4.
• \( x = -\sqrt{16} \) which equals -4.

This dual nature of square roots is essential when dealing with quadratic equations, especially when solving them, as it shows all possible solutions.

Verification of Solutions

Verification of solutions is a crucial step in solving any equation. It involves plugging the found solutions back into the original equation to ensure they hold true.
For our equation \( x^2 = 16 \), the solutions we found were \( x = 4 \) and \( x = -4 \). To verify them:

• Substitute \( x = 4 \) back into the original equation: \( 4^2 = 16 \). This checks out because 16 equals 16.
• Next, substitute \( x = -4 \): \( (-4)^2 = 16 \). Again, this is correct because squaring \(-4\) gives 16.

Through verification, both solutions satisfy the equation, confirming their correctness. Always remember, verifying solutions ensures accuracy and keeps problems error-free.