Problem 39 - Introductory Algebra for College Students

Question

Solve quadratic equation by completing the square. \(x^{2}+4 b x=5 b^{2}\)

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0\) and \(x = -4b\).

Step 1: Move the constant to the right side of the equation

You rearrange the quadratic equation \(x^{2}+4 b x=5 b^{2}\) to the form \(x^{2} + 4bx - 5b^{2} = 0\) to move the constant term (-5b^2) to one side.

Step 2: Complete the square

You complete the square on the left side of the equation. This involves finding a value that can be added to \(x^{2} + 4bx\) so that it becomes a perfect square trinomial. The value is \((b)^2\), and adding it gives \((x^{2} + 4bx + 4b^{2}) - 5b^{2} = 4b^{2}\) . The left side becomes a perfect square, \((x + 2b)^2 = 4b^{2}\).

Step 3: Solve for the variable

You take the square root of each side of the equation to solve for \(x\). This gives \(x+2b = \pm 2b\). Subtracting \(2b\) from each side of the equation results in \(x = -2b \pm 2b\).

Key Concepts

Quadratic Equations

Quadratic equations form a special category of polynomial equations, characterized by a degree of two, typically written in the form: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) represent numerical coefficients, with \( a \) never being zero, as that would reduce it to a linear equation. The term "quadratic" stems from "quad," meaning square, indicating the highest power of the variable. In solving quadratic equations, multiple strategies can be employed, such as factoring, the quadratic formula, graphing, or completing the square. The method of completing the square can often be used to convert a quadratic equation into an easily solvable format. This approach is particularly useful when the equation does not factor neatly or when precise solutions are required. Understanding the fundamental structure of quadratic equations can help in identifying the most efficient strategy for solving them.

Perfect Square Trinomial

A perfect square trinomial is a type of algebraic expression formed by squaring a binomial. It takes the form of \( (x + a)^2 = x^2 + 2ax + a^2 \). When creating a perfect square trinomial, the goal is to make the quadratic expression a square of a simple binomial. Achieving this involves the process of completing the square. Essentially, it means adjusting a quadratic expression until it matches the perfect square trinomial pattern. To transform \( x^2 + 4bx \) into a perfect square trinomial, we needed to add \( 4b^2\), turning it into \( (x + 2b)^2 \). Understanding this concept allows one to simplify complex quadratic expressions, making them easier to solve.

Solving Equations

Solving equations, particularly quadratic ones by completing the square, involves transforming the equation into a more digestible form. Once you have a perfect square trinomial, the next step is to solve for the unknown variable. In the provided example, after completing the square, we obtained \((x + 2b)^2 = 4b^2\). The solution involved finding the square roots:

• Take the square root of both sides: \( x + 2b = \pm 2b \)
• Simplify to obtain: \( x = -2b \pm 2b \)

The plus-minus symbol (±) signifies two potential solutions of the quadratic equation.This method not only reveals the solutions but also highlights the symmetry of quadratic functions, as they typically yield two solutions due to their parabolic shape when graphed. Mastery of these steps is crucial for effectively handling quadratic equations.