Problem 71 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the equation using any convenient method. $$(x+3)^{2}=81$$

Step-by-Step Solution

Answer

The solutions of the equation are \(x = 6\) and \(x = -12\).

Step 1: Simplify the square

Start by taking the square root on both sides of the equation to remove the square on the left-hand side. Doing this gives: \(x+3 = \pm \sqrt{81}\).

Step 2: Calculate the square root

The square root of 81 is 9 because \(9^2 = 81\). Substitute this into the equation to get \(x + 3 = \pm 9\). This represents two equations: \(x + 3 = 9\) and \(x + 3 = -9\).

Step 3: Solve for x

For the equation \(x + 3 = 9\), isolating \(x\) gives \(x = 9 - 3\), which simplifies to \(x = 6\). For the equation \(x + 3 = -9\), isolating \(x\) gives \(x = -9 - 3\), which simplifies to \(x = -12\).

Key Concepts

Square Roots

Square roots are essential tools in mathematics, particularly in solving quadratic equations. A square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 81 is 9 because 9 multiplied by 9 equals 81.

Understanding square roots helps in operations where squaring has been performed, as in the original equation \((x+3)^2=81\). To solve such an equation, we use the principle of square roots to eliminate the square.
This process involves taking the square root of both sides, resulting in two possible solutions:

• \(x + 3 = 9\)
• \(x + 3 = -9\)

Square roots yield both positive and negative outcomes because \((-n)^2 = n^2\). Recognizing both possibilities is key to finding all valid solutions. Always remember that each squared term will have two roots: a positive and a negative.

Isolating Variables

Isolating variables is a crucial step in solving equations, especially those involving squares. It refers to the process of manipulating an equation such that the variable you are solving for stands alone on one side of the equation.

In the equation \(x+3=9\), isolating the variable \(x\) involves subtracting 3 from both sides. This gives \(x = 9 - 3\), simplifying to \(x = 6\).

Similarly, for \(x+3=-9\), subtracting 3 from both sides results in \(x = -9 - 3\), which further simplifies to \(x = -12\).

• Make sure to perform the same operation on both sides of the equation to maintain balance.
• Simplify each step progressively to clearly identify the value of the variable.

By mastering this technique, you can effectively solve for any variable within an equation, leading to correct and simplified outcomes.

Simplifying Equations

Simplifying equations involves reducing an equation to its simplest form. This process makes it easier to understand and solve equations.

When dealing with equations like \((x+3)^2=81\), simplification begins with removing any exponents by taking square roots. This operation revealed two potential linear equations: \(x+3=9\) and \(x+3=-9\).

Further simplification brought both to the form of \(x=6\) and \(x=-12\).

• Look for opportunities to reduce the equation at each stage.
• Always work towards getting the variable on one side to simplify further.

The goal of simplification is to make equations more manageable, allowing for straightforward computation and a clear path to the solution.