Problem 94 - Elementary Algebra

Question

Solve by completing the square. \(3 q^{2}-5 q=9\)

Step-by-Step Solution

Answer

\( q = \frac{5 \pm \sqrt{133}}{6} \)

Step 1 - Move constant term to the other side

Subtract 9 from both sides to move the constant term to the right side: \[ 3q^2 - 5q - 9 = 0 \]

Step 2 - Divide by leading coefficient

Divide every term in the equation by 3 to simplify the quadratic term: \[ q^2 - \frac{5}{3}q = 3 \]

Step 3 - Complete the square

To complete the square, add and subtract the square of half the coefficient of the linear term (\( \frac{5}{3} / 2 = \frac{5}{6} \), so \( \left( \frac{5}{6} \right)^2 = \frac{25}{36} \)) to the left side of the equation: \[ q^2 - \frac{5}{3}q + \left( \frac{5}{6} \right)^2 = 3 + \frac{25}{36} \]Simplify the right side after adding \(\frac{25}{36}\): \[ q^2 - \frac{5}{3}q + \frac{25}{36} = \frac{108}{36} + \frac{25}{36} \]\[ q^2 - \frac{5}{3}q + \frac{25}{36} = \frac{133}{36} \]

Step 4 - Express the left side as a binomial square

Rewrite the left side of the equation as a perfect square: \[ \left(q - \frac{5}{6} \right)^2 = \frac{133}{36} \]

Step 5 - Solve for \( q \)

Take the square root of both sides and solve for \( q \): \[ q - \frac{5}{6} = \pm \sqrt{\frac{133}{36}} \]\[ q - \frac{5}{6} = \pm \frac{\sqrt{133}}{6} \]Add \( \frac{5}{6} \) to both sides: \[ q = \frac{5}{6} \pm \frac{\sqrt{133}}{6} \]

Step 6 - Simplify the solution

Combine the fractions: \[ q = \frac{5 \pm \sqrt{133}}{6} \]

Key Concepts

Quadratic Equations

A quadratic equation is a type of polynomial equation of the form \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are constants and \( x \) is the variable. Quadratic equations can have one, two, or no real solutions.

To solve quadratic equations, different methods can be used:

• Factoring
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Completing the square

These methods yield the same result but might be more or less convenient depending on the equation. Here, we solve the equation by completing the square.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to solve equations. Let's break down the steps in our given exercise.

Step 1: Moving the constant term to the other side

We first isolate the quadratic and linear terms by moving the constant term to the other side, resulting in: \(3q^2 - 5q - 9 = 0 \)

Step 2: Dividing by the leading coefficient

To simplify the equation, divide all terms by 3: \( q^2 - \frac{5}{3} q = 3 \)

Step 3: Completing the square

This step involves forming a perfect square trinomial on one side. We take half the coefficient of the linear term \( -\frac{5}{3}\), which is \( \frac{5}{6} \), square it to get \( \frac{25}{36} \), then add and subtract \( \frac{25}{36} \) to balance the equation: \( q^2 - \frac{5}{3} q + \frac{25}{36} = 3 + \frac{25}{36} \)

Simplify the right side: \( \frac{108}{36} + \frac{25}{36} = \frac{133}{36} \).

Step 4: Expressing as a binomial square

This step rewrites the left side as a binomial square: \( \left( q - \frac{5}{6} \right)^2 = \frac{133}{36} \).

Solving Equations

To solve the quadratic equation in its completed square form \( (q - \frac{5}{6})^2 = \frac{133}{36} \), we follow these steps:

Step 5: Solving for \( q \)

Take the square root of both sides: \( q - \frac{5}{6} = \pm \frac{\sqrt{133}}{6} \).

Step 6: Simplifying the solution

Solve for \( q \) by adding \( \frac{5}{6} \) to both sides: \( q = \frac{5}{6} \pm \frac{\sqrt{133}}{6} \).

Finally, combine the fractions into a single fraction: \( q = \frac{5 \pm \sqrt{133}}{6} \).

The solutions to the equation are \( q = \frac{5 + \sqrt{133}}{6} \) and \(q = \frac{5 - \sqrt{133}}{6} \).