Problem 33 - Algebra 2

Question

Find the two real-number solutions of each equation. $$ x^{2}=100 $$

Step-by-Step Solution

Answer

The real-number solutions of the equation \(x^{2}=100\) are \(x1=-10\) and \(x2=10\).

Step 1: Identify the problem

The equation \( x^{2}=100 \) is given. It's a type of quadratic equation in the format of \( x^{2}=a \) where a equals to 100. Therefore, we need to find two real solutions of this equation.

Step 2: Find the square root of the number on the right-hand side

The square root of 100 can be found easily. \(\sqrt{100} = 10\). The square root of a number has two results, one positive and one negative.

Step 3: Write the solutions

So, the answers to the equation \(x^{2}=100\) are \(x1 = -10\) (negative square root of 100) and \(x2 = +10\) (positive square root of 100).

Key Concepts

Real-number solutions

Quadratic equations often seek real-number solutions. These solutions are the values of the variable (usually represented as \( x \)) that satisfy the equation. Real numbers include both positive and negative numbers, zero, and fractional numbers. When dealing with quadratic equations like \( x^2 = 100 \), the task is to find all possible real solutions.

• If a quadratic equation is in the form \( x^2 = a \) where \( a \) is a positive number, it generally has two real-number solutions: one positive and one negative.
• Only positive or negative solutions indicate the direction of the real-number line they appear on.
• For equations where \( a = 0 \), the only real solution would be \( x = 0 \).
• When \( a \) is negative, real-number solutions do not exist because square roots of negative numbers are not real under typical real-number operations.

Understanding which type of solutions to expect helps simplify the process of solving quadratic equations.

Square roots

Square roots play a crucial role in solving quadratic equations. To solve an equation like \( x^2 = a \), we need the square root of \( a \). The square root operation finds a number that, when multiplied by itself, equals \( a \).

• For positive numbers, there are two possible square roots: one positive and one negative. For \( a = 100 \), the square roots are \( +10 \) and \( -10 \).
• The square root of 100 is 10, because \( 10 \times 10 = 100 \).
• The notation \( \sqrt{a} \) refers to the principal square root, which is non-negative. The negative square root is represented as \( -\sqrt{a} \).

Calculating the square root correctly determines both potential solutions to the quadratic equation. This process is essential for reaching the complete set of real-number solutions.

Solving equations

Solving equations is the process of finding values for the variable that make the equation true. For quadratic equations like \( x^2 = 100 \), the steps are straightforward yet significant.

• Identify the equation form: Recognize \( x^2 = a \) and deduce what \( a \) represents.
• Compute the square root: Find \( \sqrt{a} \) and address both the positive and negative possibilities.
• State the solutions: In our example, we find that \( x = 10 \) and \( x = -10 \) are both correct answers.

The process generally involves simplification or manipulating variables to isolate terms. In many cases, balancing the equation by performing equal operations to both sides helps maintain logical equivalence. The key is performing operations that maintain or simplify the structure of the equation, revealing real-number solutions.