Problem 43 - Algebra for College Students

Question

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$n^{2}+32 n+252=0$$

Step-by-Step Solution

Answer

The solutions are \( n = -14 \) and \( n = -18 \).

Step 1: Identify coefficients

First, identify the coefficients of the quadratic equation. The equation given is \( n^2 + 32n + 252 = 0 \). Here, \( a = 1 \), \( b = 32 \), and \( c = 252 \).

Step 2: Write the quadratic formula

The quadratic formula is given by: \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We'll use this formula to find the solutions to the equation.

Step 3: Calculate the discriminant

Calculate the discriminant \( b^2 - 4ac \). Substitute the values of \( a \), \( b \), and \( c \): \( b^2 - 4ac = 32^2 - 4 \cdot 1 \cdot 252 \). Simplifying gives \( 1024 - 1008 = 16 \).

Step 4: Solve for n using the quadratic formula

Using the quadratic formula \( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), substitute \( a = 1 \), \( b = 32 \), and \( \sqrt{16} = 4 \). Thus, \( n = \frac{-32 \pm 4}{2} \). Calculate the two possible solutions: \( n_1 = \frac{-32 + 4}{2} = -14 \) and \( n_2 = \frac{-32 - 4}{2} = -18 \).

Step 5: Verify with sum and product of roots

The sum of the roots \( n_1 + n_2 = -\frac{b}{a} = -32 \) matches \( -14 + (-18) = -32 \). The product of the roots \( n_1 \cdot n_2 = \frac{c}{a} = 252 \) matches \( (-14) \cdot (-18) = 252 \). Both checks confirm the solutions are correct.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool that helps us find the roots of any quadratic equation. A quadratic equation is typically of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants. The roots are the values of \(x\) that satisfy the equation. The quadratic formula is written as:\[n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, the symbol \(\pm\) indicates there are generally two solutions: one for the plus sign and one for the minus sign. These are known as the roots of the quadratic equation.

• Roots: Solutions to the quadratic equation.
• Quadratic formula: Provides an explicit calculation to find the roots.
• Applicability: Useful for any quadratic equation, no matter how complicated.

By substituting the values for \(a\), \(b\), and \(c\) into the formula, you can calculate the roots. Remembering this formula can make solving quadratic equations a breeze.

Discriminant

The discriminant is a component of the quadratic formula under the square root, \(\sqrt{b^2 - 4ac}\). It plays a crucial role in determining the nature and number of roots that a quadratic equation will have.

• Calculation: \(b^2 - 4ac\)
• Positive Discriminant: If the discriminant is positive, the quadratic equation has two distinct real roots.
• Zero Discriminant: If it equals zero, there is exactly one real root, or a repeated root.
• Negative Discriminant: If the discriminant is negative, there are no real roots, but instead, two complex roots.

In our example, the discriminant is 16, which is positive, indicating two distinct real roots. Understanding the discriminant allows you to predict the type of roots even before calculating them.

Sum and Product of Roots

The concepts of the sum and product of roots relate closely to the coefficients of the quadratic equation. This relationship provides a quick check to verify our calculated solutions.

• Sum of the roots: Given by \(-\frac{b}{a}\) for the equation \(ax^2 + bx + c = 0\).
• Product of the roots: Calculated as \(\frac{c}{a}\).

These relationships are derived from Vieta's formulas, which connect the coefficients of the polynomial to sums and products of its roots. They act as a sanity check after solving the quadratic equation using methods like the quadratic formula. In our example, the sum of the roots \(-14\) and \(-18\) was confirmed as \(-32\), and their product matched \(252\), verifying the solutions are indeed correct.