Problem 102 - College Algebra Essentials

Question

Exercises \(100-102\) will help you prepare for the material covered in the next section. Solve by completing the square: \(y^{2}-6 y-4=0\)

Step-by-Step Solution

Answer

The solutions to the quadratic equation are \(y = 3 + \sqrt{13}\) and \(y = 3 - \sqrt{13}\)

Step 1: Arrange the quadratic equation

Rearrange the quadratic equation in the form \(y^{2} - 6y = 4\). This format will allow easier implementation of completing the square.

Step 2: Complete the square

Add \(9\) on both sides of the equation to complete the square. This will lead to the form \((y-3)^{2} = 13\). The number \(9\) is obtained by calculating \( (6/2) ^ 2 \) .

Step 3: Solve for y

Get the square root on both sides to solve for \(y\). This will lead to two solutions, which are: \(y = 3+\sqrt{13}\) and \(y = 3-\sqrt{13}\)

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation of the form \(ax^2 + bx + c = 0\). In these equations, the highest power of the variable is squared. The challenge often is to find the values of the variable, known as "roots," which satisfy the equation.
Quadratic equations can appear in various real-world scenarios, like calculating projectile motion or optimizing profit margins. These equations are vital because they give us precise mathematical solutions to complex problems.
Understanding the standard form of a quadratic equation helps us identify how to solve it effectively. This is the form represented above: \(ax^2 + bx + c = 0\). Completing the square is a powerful technique used for solving such equations.

Solving Equations

Solving equations involves finding values that make the equation true. When it comes to quadratic equations, one elegant method is completing the square.
Completing the square transforms a quadratic equation into a perfect square trinomial, allowing for easier solving. The technique involves three pivotal steps:

• Re-arrange the equation so that the constant term is isolated on one side. This sets up the equation for the upcoming manipulation.
• Add a specific constant to both sides of the equation to convert it into a perfect square. This constant is found by taking half the coefficient of the linear term, squaring it, and adding it to both sides.
• Solve the resulting simplified equation. The equation, now a perfect square, simplifies into a form easy to solve by extracting square roots.

By carefully following these steps, you can solve quadratic equations accurately and effectively.

Algebraic Manipulation

Algebraic manipulation is the art of rearranging and simplifying equations to make them easier to solve. This includes adding, subtracting, multiplying, and dividing terms while maintaining equality. This skill is essential when solving quadratic equations.
In the process of solving the equation \(y^2 - 6y - 4 = 0\) via completing the square, one engages in several critical algebraic manipulations:

• Reorganizing terms to isolate the variable's squares and linear components.
• Adding and subtracting the same number to both sides to preserve balance.
• Using multiplication or division to simplify either side of the equation.

Algebraic manipulation demands precision and keeps the equation correctly balanced throughout the solution process. Mastering these skills is key to handling more complex algebraic problems effortlessly.