Problem 70 - Algebra 1

Question

Use a calculator to solve the equation or write no solution. Round the results to the nearest hundredth. $$6 y^{2}+22=34$$

Step-by-Step Solution

Answer

The solution to the equation is \( y = 1.41 \) and \( y = -1.41 \) when rounded to the nearest hundredth.

Step 1: Simplify the Equation

First, subtract 22 from both sides of the equation in order to isolate the quadratic term. So, we get \( 6y^2 = 34 - 22 \), which simplifies to \( 6y^2 = 12 \).

Step 2: Solve for \( y^2 \)

Next, divide each side by 6 in order to solve for \( y^2 \). That gives us \( y^2 = 12 / 6 \), which simplifies to \( y^2 = 2 \).

Step 3: Solve for \( y \)

The last step is to undo the square by taking the square root of each side. Remember when taking the square root of \( y^2 \), we need to consider both positive and negative roots. Thus, \( y = \sqrt{2} \) and \( y = -\sqrt{2} \).

Step 4: Round to the Nearest Hundredth

The last two solutions need to be rounded to the nearest hundredth. Using a calculator, the square root of 2 is approximately 1.41, and the negative square root of 2 is approximately -1.41

Key Concepts

Solving Quadratic Equations

When dealing with quadratic equations like the one given, the goal is to find the values for the variable that make the equation true. In our example, the equation is a standard quadratic form: \(6y^2 + 22 = 34\). To solve it, you first need to simplify the equation. This involves isolating the term with the variable. Typically, this is done by performing mathematical operations such as addition, subtraction, multiplication, or division.

The standard process for solving involves moving all terms to one side of the equation to set it to zero, but in simpler cases, like ours, you might directly isolate the variable term. Once it's simplified and you have an expression like \(y^2 = 2\), you can solve for \(y\) using additional mathematical techniques, like finding the square root. Solutions could be zero, one, or two, as quadratic equations can have up to two solutions due to their degree of two.

Rounding Decimals

Rounding numbers is the process of adjusting a number to be closer to a specified increment, which in this case is to the nearest hundredth. Rounding to the nearest hundredth means looking at the thousandth place (three decimal places) and determining if you should round the number up or leave it as is based on that digit.

Here’s how rounding works for our solutions. For the number \(1.414...\):

• If the thousandth digit is 5 or more, you increase the hundredth place by one — so, \(1.414\) rounds to \(1.41\).
• If the thousandth digit is less than 5, you keep the hundredth place as it is.

Negative numbers follow the same rule: \(-1.414\) also rounds to \(-1.41\). Understanding rounding is crucial because it helps make calculations simpler while maintaining a reasonable degree of accuracy.

Square Roots

Square roots are mathematical operations that find the original number (the base) which, when multiplied by itself, gives a specified value. If \(y^2 = 2\), it means you need to determine the value of \(y\) that satisfies this equation.

A square root can result in two values: positive and negative. This is due to the property of squaring where both \(-y\) and \(y\) squared will yield the same result. Thus, \(y = \sqrt{2}\) and \(y = -\sqrt{2}\) are both valid solutions. Using a calculator simplifies this process, especially when the square root does not result in a simple integer, so you can get approximate decimal values like \(\pm 1.41\).

Mathematical Operations

Mathematical operations are fundamental steps like addition, subtraction, multiplication, and division used in modifying and solving equations. In the context of solving quadratic equations, you apply these operations strategically to isolate variables and simplify expressions.

To begin, you might subtract or add numbers to both sides of the equation as needed. Following this, division and multiplication are used to isolate terms with the variable of interest time effectively. These operations are performed in accordance with mathematical conventions, often remembered using the order of operations acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right)). This ensures the solution process is logical and accurate, building the foundation for more complex algebraic manipulations.