Problem 17 - Elementary Algebra

Question

For the following problems, solve the equations by completing the square. $$ b^{2}-6 b=72 $$

Step-by-Step Solution

Answer

Question: Solve the quadratic equation \(b^2 - 6b = 72\) by completing the square method. Answer: The solutions for b are \(b = 3 + \sqrt{90}\) and \(b = 3 - \sqrt{90}\).

Step 1: Rewrite the equation

Rewrite the given equation by moving the constant term to the other side: $$ b^2 - 6b - 72 = 0 $$

Step 2: Calculate the square term

Divide the coefficient of the linear term by 2, and then square the result to find the term which will be added and subtracted to complete the square: $$ \left(\frac{-6}{2}\right)^2 = 9 $$

Step 3: Add and Subtract square term

Add and subtract the square term from the left side of the equation. Also, add the square term to the right side: $$ (b^2 - 6b + 9) - 9 - 72 = 0 + 9 $$

Step 4: Rewrite the equation in square form

Rewrite the equation in the \((b-c)^2\) form: $$ (b - 3)^2 - 81 = 9 $$

Step 5: Solve for b

Solve for b by taking square roots on both sides and simplifying: 1) Add 81 to both sides: $$ (b - 3)^2 = 90 $$ 2) Take the square root on both sides: $$ b - 3 = \pm \sqrt{90} $$ 3) Add 3 to both sides: $$ b = 3 \pm \sqrt{90} $$ So, the solutions are: $$ b = 3 + \sqrt{90} $$ and $$ b = 3 - \sqrt{90} $$

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation with a degree of 2, usually in the form of \( ax^2 + bx + c = 0 \). These equations are called 'quadratic' because 'quad' means square, referring to the squared term. In our exercise, we have \( b^2 - 6b = 72 \), which is equivalent to \( b^2 - 6b - 72 = 0 \) when we bring 72 to the other side of the equation.

Quadratic equations can have 0, 1, or 2 real solutions. They generally form a parabola when graphed on the coordinate plane. The challenge is to find the values of the variable, which make the equation true. These values are the roots of the equation.

Several methods exist to solve quadratic equations, including graphing, factoring, using the quadratic formula, and completing the square. Each method has unique advantages and is best suited for different types of problems.

• Graphing provides a visual solution.
• Factoring is simpler when equations are easily factorable.
• The quadratic formula works for any quadratic equation.
• Completing the square, as used in our solution, is particularly useful for equations that aren't easily factorable and provides a clear visual understanding of the nature of parabolas.

Algebraic Solutions

Algebraic solutions to equations refer to solving them using algebraic manipulations and techniques. In each step, we simplify or rearrange terms to find the values of the variables involved.

For quadratic equations, an algebraic solution would mean rearranging terms and completing operations like addition, subtraction, multiplication, division, or using properties like the distributive property. In our exercise, completing the square is an algebraic method.

**Steps in Algebraic Solution:**
To solve the given equation \( b^2 - 6b = 72 \), we used the following steps:

• Move the constant term to the other side of the equation to set it to zero: \( b^2 - 6b - 72 = 0 \).
• Find a number that, when added and subtracted to the left side, forms a perfect square trinomial. Here, we used \( \left(\frac{-6}{2}\right)^2 = 9 \).
• Add and subtract this number to complete the square: \( b^2 - 6b + 9 - 81 = 9 \).
• Rewrite the equation in square form: \( (b-3)^2 - 81 \).

This approach allows us to isolate the variable, making it straightforward to find its solutions.

Square Root Method

The square root method comes into play once the equation has been transformed into a perfect square form, such as \( (b-3)^2 = 90 \) in our example. This method is a straightforward approach to find the variable's value by taking the square root of both sides of the equation.

**How to Use the Square Root Method:**
When you have an equation in the form \( (x - d)^2 = e \), you can take these steps:

• First, if necessary, ensure that the equation is balanced and simplified.
• Take the square root of both sides, remembering to consider both the positive and negative roots. This results in \( x - d = \pm \sqrt{e} \).
• Solve for \( x \) by isolating it. In our example, adding 3 to both sides from \( b - 3 = \pm \sqrt{90} \) gives \( b = 3 \pm \sqrt{90} \).

The square root method is especially useful because it provides both potential solutions quickly and is a direct path to solving quadratics after completing the square. It's important to remember the ± symbol, as it reflects the two possible solutions of the quadratic equation.