Question

Solve each equation, if possible. $$ (x+7)(x-1)=(x+1)^{2} $$

Short answer

x = 2

Step-by-step solution

Expand Both Sides

Expand both sides of the equation to get rid of the parentheses. \((x+7)(x-1)\) becomes \(x^2 - x + 7x - 7\) or \(x^2 + 6x - 7\). \((x+1)^2\) becomes \(x^2 + 2x + 1\).

Set the Expanded Forms Equal

Write the expanded forms as an equation: \(x^2 + 6x - 7 = x^2 + 2x + 1\).

Subtract \(x^2\) from Both Sides

Subtract \(x^2\) from both sides to simplify: \(6x - 7 = 2x + 1\).

Isolate \(x\)

Move the \(2x\) term to the left side by subtracting \(2x\) from both sides: \(6x - 2x - 7 = 1\) or \(4x - 7 = 1\).

Solve for \(x\)

Add 7 to both sides to isolate the term with \(x\): \(4x = 8\). Divide both sides by 4 to solve for \(x\): \(x = 2\).

Verify the Solution

Substitute \(x = 2\) back into the original equation to verify: Left Side: \((2+7)(2-1) = 9\). Right Side: \((2+1)^2 = 9\). Both sides are equal, confirming \(x = 2\) is correct.

Key concepts

expanding parentheses

When solving quadratic equations, one of the initial steps often involves expanding parentheses. This process helps to simplify the equation and make it easier to work with.

• For example, in the exercise, we start with the equation \ (x+7)(x-1) = (x+1)^2 \.
• Expanding each side means removing the parentheses by distributing each term.
• For the left side, \ (x+7)(x-1) \ expands to \ x^2 - x + 7x - 7 \, which simplifies to \ x^2 + 6x - 7 \.
• For the right side, \ (x+1)^2 \ expands to \ x^2 + 2x + 1 \.

These expansions reveal the quadratic and linear terms more clearly, making the next steps in solving the equation more straightforward. The expanded form of a quadratic equation is essential for isolating variables and ultimately solving for \(x\).

isolating variables

After expanding the parentheses in a quadratic equation, the next crucial step is isolating variables. This process aims to gather all terms involving \(x\) on one side of the equation, simplifying the path to finding the value of \(x\).

• In the example, after expanding, we had \ x^2 + 6x - 7 = x^2 + 2x + 1 \.
• By subtracting \(x^2\) from both sides, the equation reduces to \ 6x - 7 = 2x + 1 \.
• Then, move the \(2x\) to the left side by subtracting \(2x\) from both sides: \ 6x - 2x - 7 = 1 \, which simplifies to \ 4x - 7 = 1 \.

Next, to isolate \(x\), add 7 to both sides to get \ 4x = 8 \. Finally, divide both sides by 4 to find \ x = 2 \. This organized method ensures that we isolate the term with \(x\) step-by-step, leading to a clearer solution.

verifying solutions

Once we have found a potential solution for \(x\), it is important to verify it by substituting it back into the original equation. Verification ensures the solution is correct.

• With \(x = 2\), substitute back into the original equation: \ (2+7)(2-1) = (2+1)^2 \.
• Simplify both sides: Left side becomes \ (9)(1) = 9 \ and Right side becomes \ (3)^2 = 9 \.
• Since both sides are equal, \ 9 = 9 \, our solution \(x = 2\) is verified as correct.

This step is vital as it confirms that the value found not only satisfies the algebraic manipulations we performed but also fits into the original quadratic equation. Verifying solutions minimizes errors and boosts confidence in the solution.