Question

Solve the equation using any method. Explain your reasoning. \(\frac{t^2}{20}+8=15\)

Short answer

Therefore, the solutions of the given equation would be \(t = \sqrt{140}\) or \(t = -\sqrt{140}\).

Step-by-step solution

Simplification

Subtract 8 from both sides of the equation, which leaves us with \(\frac{t^2}{20} = 15 - 8 = 7 \)

Isolate the Variable

To isolate 't', we first multiplicate both sides of the equation by 20, resulting in \(t^2 = 7 * 20 = 140\). Now, to eliminate the square, we take the square root of both sides which goes as follows: \(t = \sqrt{140}\) or \(t = -\sqrt{140}\). We must remember that squaring both \(+\sqrt{140}\) and \(-\sqrt{140}\) would result in the same value, which is 140, hence the result can be either of them.

Key concepts

Solving Equations

Solving equations is a fundamental skill in mathematics that involves finding the unknown variable that makes the equation true. In our problem \( \frac{t^2}{20} + 8 = 15 \), we want to discover what value of \( t \) will satisfy the equation. When solving equations, the first step is usually simplification.

• Simplification often involves performing basic arithmetic operations, such as addition, subtraction, multiplication, or division, to both sides of the equation.
• Our next step is to simplify the equation as much as possible, aiming to 'whittle down' the equation until it becomes simpler. For instance, moving constants (like the number 8 in our equation) to the other side of the equation.

This makes it easier to isolate the variable, which is the core focus in solving many algebraic equations. Once we have simplified the equation to \( \frac{t^2}{20} = 7 \), we are on the path to isolating the desired variable.

Square Root Method

The square root method is a powerful mathematical tool often used when solving equations involving squares, such as \( t^2 \) in our example equation. After simplifying and isolating the term \( t^2 \) to get \( t^2 = 140 \), the next logical step is to "unsquare" or take the square root of both sides.

• The square root method involves recognizing that squaring and taking a square root are opposite (or inverse) operations.
• When we apply the square root to both sides, we aim to make it easier to grasp what value the variable (here, \( t \)) could be.

Taking the square root of both sides yields two potential solutions: \( t = \sqrt{140} \) and \( t = -\sqrt{140} \). This dual solution arises because both the positive and negative values, when squared, yield the same result. Remember this important aspect whenever using the square root method: it usually gives two solutions because of the symmetric property of squaring numbers.

Isolating Variables

Isolating the variable is essential when solving equations because it allows us to study what specific value or values that variable can take. In mathematical equations, like our given problem, we often start by manipulating the equation to "isolate" the variable, getting it by itself on one side of the equation.

• To isolate \( t \) in our example \( \frac{t^2}{20} = 7 \), we multiply both sides by 20 to remove the fraction. This results in \( t^2 = 140 \).
• Isolating is akin to "peeling the layers" around the variable, making it as the sole item on one side of the equation. It's crucial because once isolated, we can directly solve for the variable, as in our problem where taking a square root becomes straightforward once \( t^2 \) is isolated to "t".

The end goal of isolating variables is to bring clarity and focus, rendering the problem solvable with systematic steps. By distilling the equation down to the essentials, it becomes easier to see and understand the possible answers, as demonstrated in the solution steps.