Problem 7 - Algebra 1: Concepts and Skills

Question

Determine the number of real solutions for each equation. $$ x^{2}-15=5 $$

Step-by-Step Solution

Answer

This equation has 2 real solutions.

Step 1: Simplify the equation

The equation can be simplified by adding 15 to both sides of the equation, which will get rid of the -15 from the left side. So the new equation becomes \(x^{2} = 20\).

Step 2: Solve for x

In order to get x isolated on one side of the equation, take the square root of both sides. This will result in plus and minus the square root because any number squared will always result in a positive number. So the solution for x will be \(x= \sqrt{20}\) and \(x= -\sqrt{20}\).

Step 3: Evaluate the solutions

Both \(\sqrt{20}\) and \(-\sqrt{20}\) are real numbers. Therefore, the equation \(x^{2} - 15 = 5\) has 2 real solutions.

Key Concepts

Real Solutions

Real solutions are the values of a variable that satisfy an equation in the scope of real numbers. When we talk about quadratic equations like \(x^2 - 15 = 5\), these solutions relate to the points where the curve of the equation intersects the x-axis.

Real numbers include all the positives, negatives, and zero that we can find on a number line. When solving quadratic equations, identifying whether the solutions are real or not helps us understand the nature of the curve.

• If a quadratic equation has real solutions, it means there's a tangible value or values of \(x\) that make the equation true.
• This typically happens when the expression under the square root (in the quadratic formula, for instance) is zero or positive, yielding real and not imaginary numbers.

In this example, both \( \sqrt{20} \) and \( -\sqrt{20} \) are real numbers. Thus, we confidently say the equation has two distinct real solutions.

Square Root Method

The square root method is a technique used to solve equations, especially useful when dealing with equations of the form \(x^2 = c\). In these cases, you can isolate \(x\) by taking the square root of both sides.

For our exercise, once we simplified the equation to \(x^2 = 20\), we applied the square root method to find \(x\).

Taking the square root of both sides gives:

• \( x = \sqrt{20} \)
• \( x = -\sqrt{20} \)

It is crucial to consider both the positive and negative roots because squaring either results in the same value for a given number \(c\). This is a common step in solving quadratic equations, and it helps us determine both solutions.

This method is straightforward when the variable is already squared, making it a quick way to solve quadratic problems.

Simplifying Equations

Simplifying equations is an initial step in solving most algebraic equations. It involves performing arithmetic operations to make the equation more manageable.

In our example, the equation \(x^2 - 15 = 5\) was simplified by moving constant terms to the opposite side. This step is vital because:

• It reduces the complexity of the equation, making it easier to handle.
• It isolates the variable term, aiding in further steps like taking the square root.

In this case, by adding 15 to both sides, we obtained \(x^2 = 20\). From here, direct methods like the square root method become applicable.

Simplifying isn't just about making equations look simpler; it's about preparing them for effective problem-solving techniques. Ensuring all terms are in the correct arrangement leads to fewer mistakes and quicker solutions.