Problem 5 - Algebra for College Students

Question

Solve each equation. You will need to use the factoring techniques that we discussed throughout this chapter. $$n^{2}-13 n+36=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(n = 4\) and \(n = 9\).

Step 1: Identify the Type of Quadratic

The equation given is a quadratic equation of the form \(n^2 + bn + c = 0\). We need to factor this quadratic equation to find the values of \(n\). The equation we have is \(n^2 - 13n + 36 = 0\).

Step 2: Write it in Factored Form

We are looking for two numbers that multiply to \(36\) and add up to \(-13\). After examining the factors of \(36\), we find that \(-4\) and \(-9\) multiply to \(36\) and add to \(-13\). Hence, we can write: \((n - 4)(n - 9) = 0\).

Step 3: Solve for Each Factor

To solve the equation, set each factor equal to zero: \(n - 4 = 0\) and \(n - 9 = 0\). This gives the solutions \(n = 4\) and \(n = 9\).

Step 4: Verify the Solutions

Plugging \(n = 4\) into the original equation, \(4^2 - 13 \times 4 + 36 = 0\), confirms that \(0 = 0\). Similarly, plugging \(n = 9\) into the original equation, \(9^2 - 13 \times 9 + 36 = 0\), also confirms that \(0 = 0\). Thus, both solutions are correct.

Key Concepts

Factoring Techniques

Factoring is an essential technique in solving quadratic equations. It involves expressing the quadratic equation in a product form of two binomials. In our problem, the equation is \(n^2 - 13n + 36 = 0\). The goal is to find two numbers whose product equals the constant term \(36\), and whose sum equals the linear coefficient of \(-13\).

Here's how you do it:

• Determine all possible pairs of factors of \(36\).
• Check which pair of factors adds up to \(-13\).
• For this equation, the correct numbers are \(-4\) and \(-9\) because \(-4 \times -9 = 36\) and \(-4 + -9 = -13\).

Once you have these numbers, factor the quadratic equation into \((n - 4)(n - 9) = 0\). This method is very effective for quadratics where factoring is possible. Understanding factorization helps in breaking down complex equations into simpler components.

Solving Quadratic Equations

After factoring the quadratic equation, solving it becomes straightforward. When a quadratic equation is expressed in its factored form, such as \((n - 4)(n - 9) = 0\), each factor set to zero will give us the possible solutions for the variable.

You need to solve each equation that comes from the factors:

• Set \(n - 4 = 0\), which leads to \(n = 4\).
• Set \(n - 9 = 0\), which leads to \(n = 9\).

Solving these simple linear equations gives you the roots of the original quadratic equation. For quadratic equations, the solutions represent the values of \(n\) where the equation equals zero. Always ensure these values correctly solve the original equation.

Verification of Solutions

Verifying the solutions is the final step to ensure that the factored roots are correct. This process involves substituting the roots back into the original equation to confirm that both solutions satisfy it.

Let's verify:

• Substitute \(n = 4\) into \(n^2 - 13n + 36 = 0\). Calculating gives \(16 - 52 + 36 = 0\). This confirms that \(n = 4\) is a solution.
• Substitute \(n = 9\). Calculating gives \(81 - 117 + 36 = 0\). This confirms that \(n = 9\) is a solution.

Verification is crucial to confirming that the solutions are mathematically accurate and applicable to the given problem. It ensures that no mistakes were made during factoring and solving the equation.