Problem 50 - Algebra and Trigonometry

Question

\(47-70\) 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 to the equation are \(x = \sqrt{7}\) and \(x = -\sqrt{7}\).

Step 1: Identify the equation

The given equation is a quadratic equation of the form \(x^2 - 7 = 0\). It is a simple quadratic equation without a linear term.

Step 2: Rearrange the equation

Add 7 to both sides of the equation to isolate \(x^2\) on one side: \(x^2 = 7\).

Step 3: Solve for x

Take the square root of both sides to solve for \(x\). Since square roots have two solutions, consider both the positive and negative roots: \(x = \pm \sqrt{7}\).

Step 4: Verify solutions

Plug \(x = \sqrt{7}\) and \(x = -\sqrt{7}\) back into the original equation to verify: \((\sqrt{7})^2 - 7 = 7 - 7 = 0\) and \((-\sqrt{7})^2 - 7 = 7 - 7 = 0\). Both satisfy the equation, confirming they are correct solutions.

Key Concepts

Solving Quadratic Equations

Quadratic equations are mathematical expressions that equate a polynomial of degree two to zero. In simpler terms, they usually look something like this: \( ax^2 + bx + c = 0 \). The equation provided \( x^2 - 7 = 0 \) fits this description, but you might notice it looks a bit simpler than what might be typical. It doesn't have a linear term \( bx \) or a constant term \( c \). This makes it easier to solve, using straightforward steps.

To solve any quadratic equation, one method is to rearrange the equation so that it becomes easier to apply the necessary mathematical operations, like taking square roots. After rearranging, you'll often need to isolate the variable \( x \) to find the values that satisfy the equation.

Remember, there can be either one, two, or no real solutions depending on the quadratic's discriminant. But don't worry, as the exercise shows, sometimes you'll find exactly two real solutions!

Square Root Method

The square root method is a practical and efficient technique to solve quadratic equations like \( x^2 - 7 = 0 \), where there is no linear term \( bx \). By isolating the \( x^2 \) term on one side, we can take the square root of both sides of the equation.

Here's how it works for our exercise:

• First, you isolate \( x^2 \) by adding 7 to both sides, leading to \( x^2 = 7 \).
• Next, take the square root of both sides. Remember to consider both the positive and negative roots, since squaring either makes a positive value: \( x = \pm \sqrt{7} \).

This step ensures you find all the potential real solutions the equation might have. This method is particularly handy for equations in the form \( x^2 = d \), where \( d \) is a positive number, as it directly provides the two roots by taking the square root.

Real Solutions

Real solutions are the values for \( x \) that satisfy the quadratic equation within the realm of real numbers. For the given equation \( x^2 - 7 = 0 \), we determined that \( x = \sqrt{7} \) and \( x = -\sqrt{7} \) are the real solutions.

What makes a solution 'real'? Real solutions involve numbers that we identify within the number line, meaning they aren't imaginary. Imaginary numbers include \( i \), the square root of negative one, which aren't solutions here since the discriminant (the part under the square root sign, in quadratic formula) is non-negative.

Double-checking our work is crucial. By substituting \( x = \sqrt{7} \) and \( x = -\sqrt{7} \) back into the original equation, we found both values satisfy the equation, making them indeed the real solutions.