Problem 38 - Algebra for College Students

Question

Solve each equation. Don't forget to check each of your potential solutions. \(\sqrt{n+6}=n+6\)

Step-by-Step Solution

Answer

Both \(n = -5\) and \(n = -6\) are solutions.

Step 1: Isolate the Square Root

The equation is \(\sqrt{n+6} = n+6\). The square root is already isolated, so we can proceed to the next step.

Step 2: Square Both Sides

To eliminate the square root, square both sides of the equation:\[(\sqrt{n+6})^2 = (n+6)^2\]This simplifies to:\[n + 6 = n^2 + 12n + 36\]

Step 3: Rearrange the Equation

Collect all terms on one side of the equation to form a quadratic equation:\[0 = n^2 + 12n + 36 - n - 6\]Simplify the expression:\[n^2 + 11n + 30 = 0\]

Step 4: Solve the Quadratic Equation

The quadratic equation \(n^2 + 11n + 30 = 0\) can be solved by factoring:\[(n+5)(n+6) = 0\]Set each factor to zero:1. \(n + 5 = 0\) implies \(n = -5\)2. \(n + 6 = 0\) implies \(n = -6\)

Step 5: Check Each Solution

Substitute \(n = -5\) back into the original equation:\[\sqrt{-5+6} = -5+6\]\[\sqrt{1} = 1\]This is true.Substitute \(n = -6\) into the original equation:\[\sqrt{-6+6} = -6+6\]\[\sqrt{0} = 0\]This is also true.

Key Concepts

Solving Quadratic Equations

Solving quadratic equations involves finding the value of the variable that satisfies the equation. A quadratic equation typically appears in the form \(ax^2 + bx + c = 0\). When faced with a quadratic equation, the goal is to determine the values of \(x\) that make the equation true. Here are some fundamental steps involved in solving such equations:

• Isolate the quadratic term on one side of the equation, if necessary. In our original problem, the square root was already isolated, allowing us to move directly to squaring both sides.
• After simplifying both sides, an equation of the form \(n + 6 = n^2 + 12n + 36\) emerges.
• The next step is to rearrange and simplify it to form a typical quadratic equation: \(n^2 + 11n + 30 = 0\).

This rearrangement sets the stage for solving the equation through factoring or other methods.

Factoring Quadratics

Factoring is a technique used to solve quadratic equations by expressing the equation as a product of two binomial expressions set to zero. In our problem, we are given the factorized form of the quadratic equation: \((n + 5)(n + 6) = 0\). Factoring involves the following steps:

• Identify the equation you want to factor. In this case, \(n^2 + 11n + 30 = 0\) has already been set up for factoring.
• Look for two numbers that multiply to the constant term (30) and add up to the linear coefficient (11). Here, those numbers are 5 and 6.
• Rewrite the quadratic as the product of two binomials: \((n + 5)(n + 6) = 0\).

This expression is now factored and set up to find the potential solutions for \(n\). By setting each factor equal to zero, you find the values: \(n = -5\) and \(n = -6\). Factoring simplifies the process of solving quadratics significantly by breaking it down into more manageable parts.

Checking Solutions

Once you've found potential solutions to a quadratic equation, it's crucial to check that they satisfy the original equation. This process ensures that no extraneous solutions, introduced during the solving process, are included.For both potential solutions \(n = -5\) and \(n = -6\), substitute back into the original equation \(\sqrt{n+6} = n+6\):

• For \(n = -5\):
  • Substitute into the equation: \(\sqrt{-5+6} = -5+6\), proceed to check, yielding \(1 = 1\), verifying the solution is correct.
• For \(n = -6\):
  • Substitute into the equation: \(\sqrt{-6+6} = -6+6\), proceed to check, yielding \(0 = 0\), again confirming correctness.

By substituting back, we confirm that both \(n = -5\) and \(n = -6\) satisfy the original equation. This step is an essential safeguard against errors and helps ensure the reliability of your solutions.