Problem 41 - Prealgebra

Question

Solve each equation. Round to the nearest tenth, if necessary. $$0.1 x^{2}=0.169$$

Step-by-Step Solution

Answer

The solutions are \(x = 1.3\) or \(x = -1.3\).

Step 1: Isolate the quadratic term

To solve the equation, start by isolating the term that contains the variable. Our goal is to get \(x^2\) by itself. Divide both sides of the equation by 0.1:\[ x^2 = \frac{0.169}{0.1} \] Simplify the right side:\[ x^2 = 1.69 \]

Step 2: Solve for x by taking the square root

To solve for \(x\), take the square root of both sides of the equation. Remember, taking the square root gives two solutions, the positive and negative roots.\[ x = \pm \sqrt{1.69} \] Calculate the square root of 1.69:\[ x = \pm 1.3 \]

Step 3: State the solution

Since \(x\) can be either positive or negative based on the square root operation, the solutions to the equation are:\[ x = 1.3 \quad \text{or} \quad x = -1.3 \]

Key Concepts

Solving Equations

When faced with a quadratic equation like the one in our exercise, the main aim is to find the value of the variable. Here, "solving equations" involves a few essential steps. For our problem, it began with isolating the quadratic term. This means getting the variable you are solving for onto one side of the equation by itself. In this case, since the quadratic term is multiplied by 0.1, we divide both sides by 0.1 to isolate it. This keeps the equation balanced.

Once the quadratic term is isolated, we can further solve the equation by performing operations like taking square roots or factoring. Solving quadratic equations sometimes involves additional steps or methods, but isolating the variable is the first step for simpler forms like this one.

Square Root

Taking the square root is a crucial step when dealing with quadratic equations. When you have an equation in the form of \( x^2 = a \), you can find the value of \( x \) by taking the square root of both sides.

It's important to remember that the square root operation can have two results: a positive root and a negative root. This is because both a positive number squared and its negative counterpart squared will result in the same value.

• Example: \( x^2 = 9 \) leads to \( x = \pm 3 \) because both \( 3^2 = 9 \) and \( (-3)^2 = 9 \).
• In our exercise, taking the square root of 1.69 gives us \( x = \pm 1.3 \).

Understanding this helps ensure that you are considering all possible solutions when solving the quadratic equations.

Rounding Numbers

Rounding numbers is a mathematical method used to make a number simpler and easier to use. It reduces the digits of a number while keeping its value close to what it originally was. Different situations might require different levels of precision.

In our exercise, the instruction was to "round to the nearest tenth," which focuses on one decimal place. This means looking at the number in the second decimal place to decide whether to round up or keep the number as it is.

• Example: If you have 1.34, the second decimal is 4. So, you round down to 1.3.
• When it is 1.36, the second decimal is 6. Thus, you round up to 1.4.

Such precision is particularly important in maths and science where displaying unnecessary decimal places can make results cumbersome to interpret.

Algebra

Algebra involves working with symbols and letters to represent numbers and quantities in equations and formulas. It is a fundamental part of mathematics that deals with expressing mathematical relationships. In algebra, we combine numbers and variables to solve equations or find unknowns.

For quadratics like our exercise, algebra is used to manipulate and simplify equations. Techniques such as dividing both sides by the same number, moving terms from one side of an equation to the other, and factoring are basic algebraic methods to simplify and solve equations.

• In our equation \(0.1x^2 = 0.169\), we used algebra by dividing by 0.1 to isolate the \(x^2\) term, which is a typical algebraic manipulation step.
• Algebra also helps in understanding the symmetry of equations, such as recognizing that solutions often come in pairs like positive and negative roots in square root operations.

Mastering algebra is foundational for solving not just quadratics but a wide range of mathematical problems.