Problem 28 - Algebra and Trigonometry

Question

\(9- 46\) The given equation is either linear or equivalent to a linear equation. Solve the equation. $$ (t-4)^{2}=(t+4)^{2}+32 $$

Step-by-Step Solution

Answer

The solution is \(t = -2\).

Step 1: Expand Both Sides

First, we'll expand both sides of the equation. For the left side, square the binomial: \((t-4)^2 = t^2 - 8t + 16\). For the right side, square the binomial: \((t+4)^2 = t^2 + 8t + 16\). Substitute these expansions back into the equation: \(t^2 - 8t + 16 = t^2 + 8t + 16 + 32\).

Step 2: Simplify the Equation

Now, simplify the equation by eliminating common terms and performing arithmetic operations. Cancel out \(t^2\) and 16 from both sides: \(-8t = 8t + 32\).

Step 3: Solve for t

Rearrange the equation to solve for \(t\). Subtract \(8t\) from both sides to combine like terms: \(-8t - 8t = 32\), which simplifies to \(-16t = 32\). Divide each side by \(-16\): \(t = -2\).

Key Concepts

Solving Equations

Solving equations involves finding the value of the variables in the equation, which makes the equation true. It is like solving a puzzle and usually requires a methodical approach. In linear equations, the highest power of the variable is one. That makes them simpler to solve compared to equations with higher powers.

• Start by isolating the variable on one side of the equation.
• Use inverse operations like addition, subtraction, multiplication, or division to simplify the problem.
• Always perform the same operation on both sides to maintain equality.

In this exercise, we solved the equation \( (t-4)^2 = (t+4)^2 + 32 \) by first expanding and then simplifying it. Each step brought us closer to isolating the variable, which was necessary to find its value.

Binomial Expansion

The binomial expansion is a vital technique used in algebra to transform expressions like \((t - 4)^2\) or \((t + 4)^2\) into simplified polynomial forms. This step involves applying the distributive property, also known as "foiling," which stands for "First, Outer, Inner, Last."

• Expand \((t-4)^2 = t^2 - 8t + 16\) by multiplying each term in the first binomial by each term in the second binomial.
• Similarly, \((t+4)^2 = t^2 + 8t + 16\) follows the same method.

After expansion, these quadratic expressions simplify to make following algebraic steps possible. Working through binomial expansion is crucial to breaking down complex expressions in a linear form.

Simplifying Equations

After expanding the binomials, it’s important to simplify the equations to make them easier to solve. This means making the expressions less complicated by combining like terms and getting rid of unnecessary components.

• In our equation \( t^2 - 8t + 16 = t^2 + 8t + 16 + 32 \), you can cancel identical terms on both sides, namely \( t^2 \) and 16.
• Afterwards, the equation reduces to \(-8t = 8t + 32\).

Simplification helps in clearly seeing which operations will get us to the solution quickest. It's an integral part of solving equations efficiently.

Arithmetic Operations

Arithmetic operations—such as addition, subtraction, multiplication, and division—are the building blocks of equation solving. In the context of our equation, they were used to manipulate and solve for the variable \(t\).

• We first subtracted \(8t\) from both sides: \-8t - 8t = 32\.
• This was followed by dividing each side by \(-16\) to isolate \(t\).
• The final operation revealed the solution: \(t = -2\).

Using arithmetic operations allowed us to rearrange the equation step-by-step, leading directly to finding the solution. These operations are essential for moving towards an isolated solution effectively by managing equation components with careful precision.