Problem 44 - 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. $$n^{2}-4 n-192=0$$

Step-by-Step Solution

Answer

The roots are 16 and -12.

Step 1: Identify coefficients

The given quadratic equation is in the form \( ax^2 + bx + c = 0 \). Here, \( a = 1 \), \( b = -4 \), and \( c = -192 \).

Step 2: Write the quadratic formula

The quadratic formula for finding the roots of the equation \( ax^2 + bx + c = 0 \) is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 3: Calculate the discriminant

The discriminant \( \Delta = b^2 - 4ac \) helps determine the nature of the roots. Substitute \( b = -4 \), \( a = 1 \), and \( c = -192 \) into the formula: \( (-4)^2 - 4(1)(-192) = 16 + 768 = 784 \).

Step 4: Find roots using quadratic formula

With \( \Delta = 784 \), plug it into the quadratic formula: \( x = \frac{-(-4) \pm \sqrt{784}}{2(1)} \). This simplifies to \( x = \frac{4 \pm 28}{2} \).

Step 5: Calculate the two possible roots

Solving the equations, we find \( x_1 = \frac{4 + 28}{2} = 16 \) and \( x_2 = \frac{4 - 28}{2} = -12 \).

Step 6: Verify solutions using sum and product of roots

According to Vieta's formulas, the sum of the roots \( x_1 + x_2 = -\frac{b}{a} = 4 \), and the product of the roots \( x_1 \cdot x_2 = \frac{c}{a} = -192 \). The calculated roots 16 and -12 satisfy these relationships: \( 16 + (-12) = 4 \) and \( 16 \cdot (-12) = -192 \).

Key Concepts

Discriminant

The discriminant is a key element in solving quadratic equations using the quadratic formula. It is denoted by the symbol \( \Delta \) and it provides vital information about the nature of the roots of the quadratic equation. The discriminant is calculated using the formula \( \Delta = b^2 - 4ac \).

In the given equation \( n^2 - 4n - 192 = 0 \), the discriminant is computed as \( 16 + 768 = 784 \), indicating that the roots are real and distinct because \( \Delta > 0 \).

Understanding the discriminant is crucial:

• If \( \Delta > 0 \), the quadratic equation has two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root (or a repeated root).
• If \( \Delta < 0 \), the equation has two complex (imaginary) roots.

Vieta's Formulas

Vieta's formulas provide a powerful tool for verifying the roots of quadratic equations. They relate the coefficients of the polynomial to the sum and product of its roots. For a quadratic equation in the form \( ax^2 + bx + c = 0 \):

\[ x_1 + x_2 = -\frac{b}{a} \]
\[ x_1 \cdot x_2 = \frac{c}{a} \]

In our example \( n^2 - 4n - 192 = 0 \), the formulas state:

• The sum of the roots is \( 16 + (-12) = 4 \), matching \( -\frac{b}{a} = -(-4)/1 = 4 \).
• The product of the roots is \( 16 \times (-12) = -192 \), aligning with \( \frac{c}{a} = -192/1 = -192 \).

Using Vieta's formulas is an effective way to check if the calculated roots are indeed correct.

Roots of Quadratic Equation

The roots of a quadratic equation are the values of the variable that satisfy the equation \( ax^2 + bx + c = 0 \). In other words, they are the values that make the equation equal zero.

When we use the quadratic formula, we find these roots precisely. For the equation \( n^2 - 4n - 192 = 0 \), after applying the quadratic formula, we discover the roots are \( x_1 = 16 \) and \( x_2 = -12 \).

Key insights about roots include:

• A quadratic equation can have two real roots, one real repeated root, or two complex roots.
• The nature of the roots is dictated by the discriminant. When positive, like in this problem where \( \Delta = 784 \), the roots are real and distinct.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \( x \) that satisfy the equation \( ax^2 + bx + c = 0 \). One of the most reliable methods is using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

The process involves:

• Identifying the coefficients \( a \), \( b \), and \( c \).
• Calculating the discriminant \( b^2 - 4ac \).
• Substituting these values into the formula to find \( x \).

For the given equation \( n^2 - 4n - 192 = 0 \), by substituting:

• \( a = 1 \)
• \( b = -4 \)
• \( c = -192 \)

We solve and find the roots to be \( 16 \) and \( -12 \). This structured approach helps ensure you correctly solve any quadratic equation.