Problem 75 - Algebra 1

Question

Use a calculator to solve the equation or write no solution. Round the results to the nearest hundredth. $$3 x^{2}+7=31$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 2.83\) and \(x = -2.83\).

Step 1: Isolate \(x^2\)

First, subtract 7 from both sides of the equation: \(3x^2 + 7 - 7 = 31 - 7\). This leaves you with \(3x^2 = 24\).

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

Divide both sides of the equation by 3: \(\frac{3x^2}{3} = \frac{24}{3}\). Now we are left with \(x^2 = 8\).

Step 3: Find \(x\)

Next, we need to find the square roots of 8 since \(x^2 = 8\). Remember that a square root has both a positive and a negative value. Therefore, \(x = \pm \sqrt{8}\). We then use a calculator to find the numerical values for \(\sqrt{8}\). This will give \(\pm 2.83\) (rounded to the nearest hundredth).

Key Concepts

Solving Equations

Equations are like puzzles in mathematics that need to be solved. To solve an equation involving a variable like \(x\), we perform various operations to find its value. The goal is to isolate the variable on one side of the equation. This often involves undoing whatever operation is applied to the variable. For the equation \(3x^2 + 7 = 31\), we start by removing the constant 7 from both sides. Think of it as peeling off layers until we have just \(x^2\) left. We then divide by 3 to completely isolate \(x^2\).

• Subtract constants first to simplify the equation.
• When multiplying or dividing, apply the operation to both sides to keep the equation balanced.
• Ensure that you simplify the equation step by step to avoid errors.

These steps will lead us to \(x^2 = 8\), which is a much simpler equation to work with.

Square Roots

A square root finds the original number that was squared. When we find \(x^2 = 8\), we need to discover \(x\) by finding the square roots of 8. Remember that every number has two square roots: one positive and one negative. It's like saying, "What number, when multiplied by itself, gives 8?"

• In the equation, \(x = \pm \sqrt{8}\) suggests two solutions: one positive and one negative.
• Think of \(\sqrt{8}\) as a mathematical operation to undo squaring.
• Use a calculator to get an approximate value for \(\sqrt{8}\); in this case, it's about 2.83.

Square roots are fundamental to solving quadratic equations and appear often in real-world applications.

Rounding Numbers

Rounding numbers is crucial when you need to simplify calculations and communicate results more succinctly. When asked to round to the nearest hundredth, you keep two digits after the decimal point. This often involves looking at the third decimal digit to decide whether to round up or stay the same. Consider \(\sqrt{8} \approx 2.828427\). To round to the nearest hundredth:

• Look at the digit in the thousandth place, here it's 8.
• Since 8 is greater than 5, we round the second decimal place up, making it 2.83.
• Rounding helps provide precise yet concise results in both academic and practical scenarios.

This approach simplifies values without losing significance, making it easier to understand and apply the numbers in further calculations.