Problem 55 - Algebra 1: Concepts and Skills

Question

Use a calculator to solve the equation. Round the result to the nearest hundredth. $$ 5 x^{2}-12=5 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(x \approx 1.84\) and \(x \approx -1.84\).

Step 1: Add 12 to Both Sides

Start by adding 12 to both sides of the equation to isolate the term with x: \(5x^{2} = 5+12\). This simplifies to \(5x^{2} = 17\).

Step 2: Divide by 5

Divide both sides of the equation by 5 to solve for \(x^{2}\): \(\frac{5x^{2}}{5} = \frac{17}{5}\). After the division, this simplifies to \(x^{2} = 3.4\).

Step 3: Take the Square Root

To solve for x, take the square root of both sides of the equation: \(x = \sqrt{3.4}\). Be aware that both the positive and negative square root should be considered in the solution.

Step 4: Calculate and Round

Using a calculator, the positive and negative square root of 3.4 round to the nearest hundredth results in \(x \approx ±1.84\).

Key Concepts

Algebraic Manipulation

Algebraic manipulation is essential in solving quadratic equations. It involves rearranging the equation to make the variable of interest the subject of the formula. This is done through various algebraic operations such as addition, subtraction, multiplication, division, and factoring.

For instance, in the exercise \(5x^2 - 12 = 5\), we start by adding 12 to both sides, an operation known as 'balancing the equation,' to isolate the quadratic term. By doing this, the equation simplifies to \(5x^2 = 17\). The next step is to divide every term by 5, which is another form of manipulation that simplifies the quadratic term to \(x^2 = 3.4\). This process demonstrates algebraic manipulation's role in setting the stage for further steps to solve the equation.

Square Roots

Square roots are profound in solving quadratic equations, especially when the equation is simplified to the form \(x^2 = c\), where \(c\) is a constant. Taking the square root is the inverse operation of squaring and is necessary for finding the value of \(x\).

In the given exercise, once we have \(x^2 = 3.4\), we apply the square root operation to both sides to find the values of \(x\). It's critical to remember that a squared variable can have both positive and negative solutions because squaring either a positive or negative number will yield the same positive result. Therefore, the equation \(x^2 = 3.4\) leads to \(x \approx ±\text{sqrt}(3.4)\), meaning we consider both \(x \approx 1.84\) and \(x \approx -1.84\) as potential solutions.

Using Calculators in Algebra

Calculators are very useful when it comes to solving algebraic equations, especially for operations that are tedious or complex to perform manually, such as finding square roots of non-perfect squares.

In our exercise, after algebraic manipulation and square root application, the final step is to use a calculator. With \(x^2 = 3.4\), we use the calculator to find the square root of 3.4. Since most calculators have a dedicated square root function, this calculation is straightforward. The calculator gives a more precise value which we then round to the nearest hundredth, obtaining \(x \approx ±1.84\). Utilizing calculators in algebra streamlines the solving process and helps verify our solutions.