Problem 20 - College Algebra with Corequisite Support

Question

For the following exercises, solve the quadratic equation by using the square root property. $$ x^{2}=49 $$

Step-by-Step Solution

Answer

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

Step 1: Understand the Square Root Property

The square root property states that if \( x^2 = a \), then \( x = \pm \sqrt{a} \). This means that to solve the equation \( x^2 = a \) for \( x \), you can find both positive and negative square roots of \( a \).

Step 2: Identify the Value of 'a'

In the equation \( x^2 = 49 \), you identify that the value of \( a \) is 49. We will use this value to apply the square root property in the next step.

Step 3: Apply the Square Root Property

Apply the square root property to \( x^2 = 49 \):\[ x = \pm \sqrt{49} \]

Step 4: Calculate the Square Roots

Find the square root of 49. \( \sqrt{49} = 7 \)Therefore, using the square root property we have:\[ x = \pm 7 \]

Step 5: Write the Solutions

The solutions to the equation \( x^2 = 49 \) are \( x = 7 \) and \( x = -7 \). These are the two values of \( x \) that satisfy the original quadratic equation.

Key Concepts

Square Root Property

The square root property is a handy tool when dealing with quadratic equations, especially those in the simple form of \( x^2 = a \). This property helps to quickly find the values of \( x \) without going through more complicated factoring or formula-based methods.

Here's how it works: If you have an equation like \( x^2 = a \), you can solve it by applying the square root to both sides. This gives you \( x = \pm \sqrt{a} \). The plus-minus sign is crucial as it indicates that both the positive and the negative roots are solutions.

For example, let's use the equation \( x^2 = 49 \). When you apply the square root property, you get \( x = \pm \sqrt{49} \). Since \( \sqrt{49} = 7 \), the solutions are \( x = 7 \) and \( x = -7 \). These are the numbers that satisfy the equation when squared, demonstrating the practical use of the square root property.

Solving Quadratics

Quadratic equations can often seem daunting, but they become simpler to solve with methods like the square root property.

A quadratic equation typically has the form \( ax^2 + bx + c = 0 \). In certain cases, like with \( x^2 = a \), the square root method is particularly efficient. It becomes an excellent choice when there's no linear term (the \( bx \) part is missing), making the solution process straightforward.

To use the square root property to solve a quadratic equation:

• Make sure the equation is in the form \( x^2 = a \).
• Identify the value of \( a \).
• Apply the square root property by taking both positive and negative square roots.
• Solve for \( x \) to get \( x = \pm \sqrt{a} \).

By following these steps, solving quadratics becomes a more manageable process.

Algebra

Algebra is a branch of mathematics that deals with symbols and rules for manipulating those symbols. It is the foundational tool used for solving equations like quadratics.

In algebra, you often work with equations to find values that make these calculations true. This involves variables, constants, and coefficients. Quadratic equations, like \( x^2 = 49 \), are a stepping stone to understanding more complex algebraic structures.

When solving any equation, especially quadratics:

• Understand the format of your equation and identify its components.
• Apply appropriate methods—such as factoring, completing the square, or using the square root property.
• Verify your solutions by substituting them back into the original equation.

Mastering these concepts in algebra not only helps in solving specific problems but also builds a strong mathematical foundation for more advanced studies.