Problem 47 - Algebra and Trigonometry

Question

\(47-70\) The given equation involves a power of the variable. Find all real solutions of the equation. $$ x^{2}=49 $$

Step-by-Step Solution

Answer

The solutions are \(x = 7\) and \(x = -7\).

Step 1: Identify the Equation

The equation given is \(x^2 = 49\). This is a quadratic equation where we need to find all real solutions for \(x\).

Step 2: Apply the Square Root Property

To solve for \(x\), take the square root of both sides of the equation. The square root property states that if \(x^2 = a\), then \(x = \pm \sqrt{a}\).

Step 3: Calculate the Square Root Values

Applying the square root property to \(x^2 = 49\), we get \(x = \pm \sqrt{49}\). Since \(\sqrt{49} = 7\), both positive and negative solutions are considered: \(x = 7\) and \(x = -7\).

Step 4: Verify the Solutions

Confirm that both \(x = 7\) and \(x = -7\) satisfy the original equation. Substituting back: \(7^2 = 49\) and \((-7)^2 = 49\), which verifies both solutions.

Key Concepts

Understanding the Square Root Property

When faced with a quadratic equation like \(x^2 = 49\), a powerful tool to use is the square root property. This property helps simplify situations where you have an unknown variable raised to the second power, such as \(x^2\). The square root property tells us that if you have an equation like \(x^2 = a\), then the solutions for \(x\) will be \(x = \pm \sqrt{a}\). This means you take both the positive and negative square roots of \(a\). Why do we need both? Because squaring either a positive or a negative number results in a positive outcome. Hence, for an equation \(x^2 = 49\), the solutions aren't just \(7\), but also \(-7\). By acknowledging this property, you cover all possibilities laid out by the quadratic equation.

Exploring Real Solutions in Quadratics

Real solutions to an equation are numbers that can actually be plotted on a number line. For a quadratic equation like \(x^2 = 49\), it translates to figuring out which real numbers, when squared, equal 49. Using the square root property, we find two real solutions: \(x = 7\) and \(x = -7\). This means both these numbers can be squared to give 49.

• The term 'real' indicates that these solutions do not involve imaginary numbers (like \(i = \sqrt{-1}\), which would arise in trying to take the square root of negative numbers).
• Real solutions are particularly valuable because they reflect quantities or measurements we encounter in day-to-day life.

Quadratic equations can often yield two real solutions, no solutions, or occasionally, complex numbers if they are more complicated.

Verifying Quadratic Solutions

Verification is an essential step in solving equations to ensure the solutions are correct. For the equation \(x^2 = 49\), once we propose the solutions \(x = 7\) and \(x = -7\), it's important to plug them back into the original equation to check their validity. Verification involves:

• Inserting \(7\) back in: \(7^2 = 49\), which holds true.
• Inserting \(-7\) back in: \((-7)^2 = 49\), also holds true.

Both calculations confirm that the proposed solutions satisfy the original equation. Far from redundant, verifying ensures that no mistakes happened in earlier steps and that the solutions are accurate. This not only boosts confidence in the solution but demonstrates a thorough understanding of the quadratic equation solving process.