Problem 32 - Intermediate Algebra

Question

Use the method of completing the square to solve each quadratic equation. $$ y^{2}-9 y+30=0 $$

Step-by-Step Solution

Answer

The solutions are \( y = 4.5 \pm i\sqrt{9.75} \).

Step 1: Identify the Quadratic Equation

The given quadratic equation is \( y^{2} - 9y + 30 = 0 \). Our goal is to solve for \( y \) by completing the square.

Step 2: Move Constant to the Other Side

We first rearrange the equation to isolate the constant term on the right side. Subtract 30 from both sides:\[ y^{2} - 9y = -30 \]

Step 3: Find the Completing Term

Identify the coefficient of \( y \), which is \(-9\). Divide this coefficient by 2 and square it: \(-9/2 = -4.5\) and \((-4.5)^2 = 20.25\). Add this term to both sides of the equation:\[ y^{2} - 9y + 20.25 = -30 + 20.25 \]

Step 4: Simplify Both Sides

Simplify the right side by adding the constant:\[ y^{2} - 9y + 20.25 = -9.75 \]

Step 5: Write Left Side as a Perfect Square

The left side is now a perfect square trinomial. Express it as the square of a binomial:\[ (y - 4.5)^{2} = -9.75 \]

Step 6: Solve for y

Take the square root of both sides:\[ y - 4.5 = \pm \sqrt{-9.75} \]Since the right side involves the square root of a negative number, the solutions will be complex. Solving further:\[ y - 4.5 = \pm i\sqrt{9.75} \]Finally, solve for \( y \):\[ y = 4.5 \pm i\sqrt{9.75} \]

Step 7: Write the Final Complex Solutions

The solutions to the equation \( y^{2} - 9y + 30 = 0 \) are:\[ y = 4.5 + i\sqrt{9.75} \]\[ y = 4.5 - i\sqrt{9.75} \]

Key Concepts

Understanding Quadratic Equations

In mathematics, a quadratic equation is any equation that can be rearranged in standard form as \( ax^2 + bx + c = 0 \), where \( x \) represents an unknown, and \( a, b, \) and \( c \) are constants. Here, \( a eq 0 \) because if \( a = 0 \), the equation simplifies to a linear equation. Quadratic equations are fundamental in algebra and are widely used to find values of variables where squared terms are involved.

To solve a quadratic equation, we often use techniques like factoring, the quadratic formula, or completing the square, which provides a way to rewrite the equation more simply and find solutions for \( x \). Understanding these methods will help tackle a wide range of problems.

Some quadratic equations have solutions that are integers or real numbers, while others have complex solutions when they involve non-real numbers. A quick check for the type of solutions involves calculating the discriminant \( b^2 - 4ac \). If it is negative, the solutions will be complex numbers.

Discovering Complex Solutions

Complex solutions arise when we encounter the square roots of negative numbers during an equation-solving process. Normally, the square root of a negative number cannot be computed within the set of real numbers. This is where complex numbers come into play.

A complex number is generally expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit with the property that \( i^2 = -1 \). When solving a quadratic equation, if we find that the discriminant \( b^2 - 4ac \) is less than zero, it's an indicator of complex solutions.

In such cases, as in the example provided, completing the square might lead to square roots of negative numbers and hence complex solutions are formed. This is illustrated by solutions like \( y = 4.5 \pm i\sqrt{9.75} \), where \( \pm i \) signifies that the solutions include both the positive and negative imaginary components.

Perfect Square Trinomial and Completing the Square

A perfect square trinomial is a special quadratic expression of the form \((x + a)^2\) or \((x - a)^2\). These trinomials expand to \(x^2 + 2ax + a^2\) or \(x^2 - 2ax + a^2\), respectively. Recognizing these helps simplify quadratic equations, making them easier to solve.

Completing the square is a technique used to transform a quadratic equation into a form that includes a perfect square trinomial. Here’s the process:

• Start by isolating the terms with the variable on one side of the equation.
• Take half of the coefficient of the linear term, square it, and add this square to both sides of the equation.
• As a result, one side of the equation becomes a perfect square trinomial, which can be expressed as a squared binomial.

In the example equation \( y^2 - 9y + 30 = 0 \), we complete the square by focusing on the linear term \(-9y\), adding \(20.25\) (the square of half of \(-9\)), leading to the trinomial \( (y - 4.5)^2 \). This step is crucial as it simplifies further solving, especially when dealing with complex solutions.