Problem 20 - Algebra 1: Concepts and Skills

Question

Solve the equation by cross multiplying. Check your solutions. \(\frac{(x+1)^{2}}{(x-3)^{2}}=1\)

Step-by-Step Solution

Answer

The solution to the equation \(\frac{(x+1)^{2}}{(x-3)^{2}}=1\) is \(x = 1\).

Step 1: Cross-multiplication

Cross multiplication in a proportion is applied by multiplying the denominator of the first ratio by the numerator of the second ratio and equates it to the product of the numerator of the first ratio and the denominator of the second ratio. Apply this concept to the equation \(\frac{(x+1)^{2}}{(x-3)^{2}}=1\). Written in terms of cross multiplication, this becomes \((x+1)^{2}*1=(x-3)^{2}*1\). Simplified, this gives: \((x+1)^{2}=(x-3)^{2}\)

Step 2: Simplify the equation

Expand the square terms on both sides of the equation. This leads to \(x^{2} + 2x + 1 = x^{2} - 6x + 9\). Then simplify the equation by subtracting \(x^{2}\) and \(2x\) from both sides of the equation. This reduces the equation to \(0 = -8x + 8\)

Step 3: Solve for x

Solving for x involves isolating x on one side of the equation. This can be done by adding 8x and subtracting 8 from both sides, which transforms the equation into \(8x = 8\). Divide both sides of the equation by 8 to find the value of x, hence \(x = 1\).

Step 4: Verify the solution

Check if the value of x found is correct by plugging it into the original equation \(\frac{(x+1)^{2}}{(x-3)^{2}}=1\). The left-hand side of the equation becomes \(\frac{(1+1)^{2}}{(1-3)^{2}}\) which simplifies to \(\frac{4}{4}\) or 1. Since this is equal to the right-hand side of the equation, the solution \(x = 1\) is valid and correct.

Key Concepts

Cross Multiplication

When you encounter a proportion, such as \(\frac{a}{b} = \frac{c}{d}\), cross multiplication is a nifty tool. It helps us determine if the ratios are equal and simplifies finding unknowns. Here's how it works:

• Multiply across the diagonals, so you compute \(a \cdot d = b \cdot c\).
• This essentially balances the equation using the cross products.

Using cross multiplication, we transformed the original problem \(\frac{(x+1)^2}{(x-3)^2} = 1\) into \((x+1)^2 = (x-3)^2\). This makes solving much easier by getting rid of the denominator.

Proportions

Proportions express that two ratios or fractions are equivalent. They are written in the form \(\frac{a}{b} = \frac{c}{d}\). Understanding proportions is crucial because they appear in various mathematical contexts, from algebra to geometry. To handle proportions:

• Ensure both sides are ratios.
• You can set them equal and cross multiply to solve for unknowns.

By setting two equal expressions, you can solve for the variable with relative ease. This is particularly helpful in solving equations like the one in the exercise, where you need to balance and simplify.

Quadratic Equations

Quadratic equations are those that can be written in the form \(ax^2 + bx + c = 0\). They often have two solutions because they represent a parabola when graphed. Solving these equations involves several methods, including factoring, completing the square, and using the quadratic formula.Breaking down the solution process:

• Expand and simplify the expression, here \((x+1)^2 = (x-3)^2\) became \(x^2 + 2x + 1 = x^2 - 6x + 9\).
• Simplify by collecting like terms, which led to the linear equation \(-8x + 8 = 0\).
• Finally, solve for \(x\) by isolating the variable, resulting in \(x = 1\).

Quadratic equations, like the one you find in this exercise, can often be simplified to first-degree equations, resulting in a straightforward solution pathway. Understanding how to manipulate and solve these can vastly improve problem-solving skills in mathematics.