Problem 54 - Algebra and Trigonometry

Question

The given equation involves a power of the variable. Find all real solutions of the equation. \(x^{2}-7=0\)

Step-by-Step Solution

Answer

The real solutions are \(x = \sqrt{7}\) and \(x = -\sqrt{7}\).

Step 1: Identify the Equation

The given equation is a quadratic equation, and it is written as: \(x^{2}-7=0\). Our goal is to find all real solutions for \(x\).

Step 2: Isolate the Quadratic Term

To solve for \(x\), first isolate the \(x^2\) term by adding 7 to both sides of the equation: \(x^{2} = 7\).

Step 3: Solve for x

To find the values of \(x\), take the square root of both sides. This gives us two possible solutions: \(x = \pm\sqrt{7}\).

Step 4: Confirm the Solutions

Verify the solutions by substituting them back into the original equation. For \(x = \sqrt{7}\): \((\sqrt{7})^2 = 7\). For \(x = -\sqrt{7}\): \((-\sqrt{7})^2 = 7\). Both satisfy the original equation.

Key Concepts

Solving Quadratics

Quadratic equations are expressions that follow the pattern: \( ax^2 + bx + c = 0 \). In these equations, \( a \), \( b \), and \( c \) are constants, with the variable \( x \) being raised to the second power or squared. Solving a quadratic equation involves finding the value of \( x \) that makes the equation true, known as the roots or solutions of the equation.
To solve, we often follow these steps:

• Identify the quadratic equation and ensure all terms are on one side.
• Isolate the quadratic term by performing algebraic operations.
• Use methods like factoring, completing the square, or the quadratic formula to solve for \( x \).

The equation \( x^2 - 7 = 0 \) is already in a simplified form and can be solved easily by isolating the \( x^2 \) term. Once isolated, the next step often involves taking square roots to find the solution(s).

Square Roots

Taking square roots is a fundamental concept in solving quadratic equations, especially ones like \( x^2 = n \). When you "take the square root" of both sides of an equation, you're essentially finding a value that, when multiplied by itself, gives the desired number.
For example, if you have \( x^2 = 7 \), the square root of 7 will provide the solutions, \( x = \pm \sqrt{7} \).

• The symbol \( \pm \) is important here. It indicates there are two possible solutions: one positive and one negative. This is because both \( (\sqrt{7})^2 \) and \( (-\sqrt{7})^2 \) equal 7.
• Remember that the square root process introduces these two potential answers because a square is always non-negative, so both positive and negative roots are considered.

Understanding and applying the concept of square roots is crucial when solving quadratics directly through equating and square rooting as shown in the example.

Real Solutions

In the context of quadratic equations, real solutions refer to the values of \( x \) that are real numbers. Not all quadratic equations have real solutions; sometimes the roots are complex. However, when solving equations like \( x^2 = 7 \), the solutions are real because \( \sqrt{7} \) is a real number.

• Real numbers include all the numbers on the number line: positive numbers, negative numbers, zero, and fractions or irrational numbers like \( \sqrt{7} \).
• In our specific equation \( x^2 - 7 = 0 \), we obtain real solutions because we're dealing with positive quantities (squaring a number and setting it equal to a positive value).

Ensuring that we correctly identify whether solutions are real or complex is a key component of solving quadratic equations and understanding their scope. Verifying the solutions by substitution is a practical way to confirm they satisfy the original equation, as was done in the exercise, confirming real solutions \( x = \pm \sqrt{7} \).