Problem 33 - Elementary Algebra

Question

For the following problems, solve the equations by completing the square. $$ y^{2}-3 y+10=0 $$

Step-by-Step Solution

Answer

Answer: The solutions are \(y = \frac{3}{2} + \sqrt{-\frac{31}{4}}\) and \(y = \frac{3}{2} - \sqrt{-\frac{31}{4}}\).

Step 1: Rewrite the equation in the form of \((y - a)^2 = b\)

First, we will rewrite the given equation so that it is in the form \((y - a)^2 = b\). Given equation is: $$ y^{2} - 3y + 10 = 0 $$ Subtract 10 from both sides: $$ y^{2} -3y = -10 $$

Step 2: Identify the values of \(a\) and \(b\)

To complete the square, we must insert a term into the left-hand side of the equation that allows us to factor it into a perfect square trinomial. This term will be \((\frac{b_{1}}{2})^2\), where \(b_{1}\) is the coefficient of the linear term in the equation (in this case, -3). Calculate this value: $$ \Big(\frac{-3}{2}\Big)^{2} = \Big(\frac{3}{2}\Big)^{2} = \frac{9}{4} $$ Now add this value to both sides of the equation to complete the square: $$ y^{2} - 3y + \frac{9}{4} = -10 + \frac{9}{4} $$ Factor the left-hand side and simplify the right-hand side: $$ (y - \frac{3}{2})^{2} = -\frac{31}{4} $$

Step 3: Use the square root property to isolate y

Using the square root property, take the square root of both sides of the equation. Remember that when you take the square root of both sides, there will be two different solutions. $$ y - \frac{3}{2} = \pm\sqrt{-\frac{31}{4}} $$

Step 4: Write the final answer

Now, add \(\frac{3}{2}\) to both sides to solve for \(y\): $$ y = \frac{3}{2} \pm \sqrt{-\frac{31}{4}} $$ So the solutions to the given equation are: $$ y = \frac{3}{2} + \sqrt{-\frac{31}{4}} \quad \text{and} \quad y = \frac{3}{2} - \sqrt{-\frac{31}{4}} $$

Key Concepts

Quadratic Equations

Quadratic equations are an essential part of algebra, often characterized by their degree, which is 2. This means the highest power of the variable, typically represented as \(x\) or another letter like \(y\), is squared. A general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants.
Solving quadratic equations can be accomplished through various methods, such as factorization, the quadratic formula, or completing the square. Each method offers different insights and solutions, particularly when an equation does not factor neatly.
When solving quadratic equations by completing the square, the goal is to manipulate the equation so that one side forms a perfect square trinomial. This technique simplifies solving for the variable, especially when the equation doesn’t easily lend itself to simpler factoring methods.

Perfect Square Trinomial

A perfect square trinomial is a quadratic expression that results in a squared binomial. It takes the form \((x - h)^2 = x^2 - 2hx + h^2\). Completing the square transforms a quadratic equation into this form, making it easier to solve.
In the process of completing the square, an additional term is often required to achieve a perfect square trinomial. For example, given \(y^2 - 3y\), we add \(\left(\frac{-3}{2}\right)^2 = \frac{9}{4}\) to perfectly square the trinomial as \((y - \frac{3}{2})^2\).
By converting the equation into a perfect square trinomial, the task of finding the value of the variable becomes more straightforward, as it can be solved using the square root property.

Square Root Property

The square root property is an algebraic tool that simplifies solving equations of the form \((expression)^2 = k\). The square root property tells us that if \((expression)^2 = k\), then \(expression = \pm \sqrt{k}\).
Applying the square root property after forming a perfect square trinomial lets you isolate the variable with ease. In the equation \((y - \frac{3}{2})^2 = -\frac{31}{4}\), applying the square root property gives \(y - \frac{3}{2} = \pm \sqrt{-\frac{31}{4}}\).
This quick method immediately gives two potential solutions, which are then adjusted by further algebraic operations to uncover the final values for the variable.

Complex Solutions

Complex solutions arise when the attempt to solve a quadratic equation involves taking the square root of a negative number. In our equation, \((y - \frac{3}{2})^2 = -\frac{31}{4}\), the negative under the square root signals that the solutions will include imaginary numbers.
In these cases, the square root of a negative number is represented using the imaginary unit \(i\), where \(i\) equals the square root of -1. The solution, therefore, becomes \( y = \frac{3}{2} \pm i\sqrt{\frac{31}{4}} \).
Understanding that complex solutions are legitimate solutions broadens your mathematical toolkit. Complex solutions are crucial in fields such as engineering and physics, where they help represent and solve real-world phenomena that involve waveforms or oscillations.