Problem 31 - Elementary Algebra

Question

Solve by completing the square. $$y 2-20 y=-25$$

Step-by-Step Solution

Answer

The solutions are \( y = 10 + 5\sqrt{3} \) and \( y = 10 - 5\sqrt{3} \).

Step 1: Move Constant to the Right

We start by isolating the terms with the variable on one side. The given equation is \( y^2 - 20y = -25 \). First, add 25 to both sides to obtain: \( y^2 - 20y + 25 = 0 \).

Step 2: Complete the Square

To complete the square, take the coefficient of the linear term (\(-20\)), halve it, then square it. Half of \(-20\) is \(-10\), and squaring \(-10\) gives \(100\). Add and subtract \(100\) in the equation to keep it balanced: \( y^2 - 20y + 100 - 100 + 25 = 0 \).

Step 3: Simplify and Rearrange

Combine like terms: \( y^2 - 20y + 100 - 75 = 0 \). This can be written as \( (y - 10)^2 - 75 = 0 \).

Step 4: Solve for y

Add 75 to both sides to get \( (y - 10)^2 = 75 \). Take the square root of both sides to get \( y - 10 = \pm \sqrt{75} \).

Step 5: Simplify the Square Root

Simplify \( \sqrt{75} \) to get \( 5\sqrt{3} \), yielding \( y - 10 = \pm 5\sqrt{3} \).

Step 6: Isolate y

Add 10 to both sides to solve for \( y \). This yields \( y = 10 \pm 5\sqrt{3} \).

Key Concepts

Quadratic Equations

Quadratic equations are fundamental in algebra and appear in the form \( ax^2 + bx + c = 0 \). They are characterized by the highest power of the variable being two. In these equations, \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \) as the distinguishing feature of quadratic equations.
Quadratic equations can model a wide array of real-world scenarios, such as projectile motion or calculating areas. Understanding how to solve these equations is crucial for students to advance in algebra and towards more complex mathematics concepts like calculus.

Solving Equations

When solving equations, the main goal is to find the value or values of the variable that make the equation true. Solving equations can involve several methods, each with their specific uses and advantages.
For quadratic equations, common methods include:

• Factoring, which is useful when the equation can be expressed as a product of binomials.
• The quadratic formula, which provides a straightforward way to find solutions.
• Graphing, which helps visualize where the equation equals zero.
• Completing the square, a method that transforms a quadratic equation into a perfect square trinomial.

Each of these methods provides a different perspective, and often selecting the best method depends on the specific form and coefficients of the quadratic equation at hand.

Algebraic Methods

Algebraic methods are systematic techniques used to manipulate equations and expressions to solve or simplify them. Completing the square is a powerful algebraic method particularly useful for solving quadratic equations.
The process involves:

• Rewriting the equation in the format \( ax^2 + bx = -c \).
• Halving the coefficient of \( x \), then squaring it, to create a perfect square trinomial.
• Adding and subtracting this square in the equation to keep it balanced.

This method continuously proves to be beneficial as it also lends itself to solving by graphing and forms the basis for deriving the quadratic formula.