Problem 50 - Algebra 1: Concepts and Skills

Question

Use a calculator to solve the equation. Round the result to the nearest hundredth. $$ 4 x^{2}-3=57 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = +3.87\) and \(x = -3.87\).

Step 1: Rearrange the equation

To make the equation easier to solve, first add 3 to both sides to isolate the term involving x on one side of the equation. This results in \(4x^2 = 60\).

Step 2: Solve for x^2

Next, divide both sides of the equation by 4 to solve for \(x^2\). This results in \(x^2 = 15\).

Step 3: Solve for x

Now take the square root of both sides to solve for x. Note that a square root has two solutions, one positive and one negative. Round the results to the nearest hundredth to get \(x = +3.87\) and \(x = -3.87\).

Key Concepts

Solving Equations

When solving equations, the goal is to find the value(s) of the variable that make the equation true. In the case of quadratic equations like the one you're dealing with, you want to isolate the variable, step by step.

• Initial step is to rearrange the equation so that the terms involving the variable are on one side. This means you might need to add, subtract, multiply, or divide both sides of the equation.
• Next, simplify the equation as much as possible. For example, in \(4x^2 - 3 = 57\), adding 3 to both sides simplifies it to \(4x^2 = 60\).
• Further simplification involves dividing both sides by 4 to isolate \(x^2\), resulting in \(x^2 = 15\).

Rearranging and simplifying the equation step by step transforms a complex problem into a simpler one, allowing you to find the solutions with ease.

Square Roots

Taking square roots is a crucial part of solving quadratic equations. Once you have an equation in the form \(x^2 = value\), solving for \(x\) requires extracting the square root:

• Remember, every positive number has two square roots: one positive and one negative. For example, while solving \(x^2 = 15\), we find \(x = \sqrt{15}\) and \(x = -\sqrt{15}\).
• This step is vital because it acknowledges that quadratic equations often have two roots, which are the solutions to the equation.
• In the exercise, once you find the square root of 15 using a calculator, the results are \(+3.87\) and \(-3.87\). This means both values satisfy the equation \(x^2 = 15\).

Understanding square roots helps in grasping why quadratic equations have two solutions and assists in solving other similar problems.

Rounding Numbers

Rounding numbers is often necessary to present results in a more understandable manner, especially in math problems where solutions require precision only to a certain point.

• In most contexts, rounding to the nearest hundredth means keeping two decimal places in your answer. So, if you get \(x = 3.872983346\), rounding it gives \(x = 3.87\).
• Rounding rules: If the digit after your rounding place is 5 or more, round up. If it's 4 or less, keep it the same. This preserves the accuracy of your result to the desired place value.
• In the solution given, both \(+3.87\) and \(-3.87\) are rounded results, ensuring your answers are easy to comprehend and neatly presentable.

Rounding numbers keeps solutions concise, accurate to the needed decimal places and interpretable, which is essential for appreciating the precise nature of mathematics.