Problem 105 - Elementary Algebra

Question

Solve by using the Quadratic Formula. \(r^{2}-8 r-33=0\)

Step-by-Step Solution

Answer

The solutions are \(r = 11\) and \(r = -3\).

Step 1: Identify coefficients

The first step is to identify the coefficients in the quadratic equation. In the equation \[ r^{2}-8r-33=0 \]the coefficient of \(r^2\) is \(a = 1\), the coefficient of \(r\) is \(b = -8\), and the constant term is \(c = -33\).

Step 2: Write down the Quadratic Formula

Recall the Quadratic Formula given by:\[ r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Now, substitute the values of \(a\), \(b\), and \(c\) into the formula.

Step 3: Calculate the discriminant

First, calculate the discriminant \(b^2 - 4ac\):\[ (-8)^2 - 4(1)(-33) = 64 + 132 = 196 \]The discriminant is 196.

Step 4: Compute the square root of the discriminant

Find the square root of the discriminant 196:\[ \sqrt{196} = 14 \]

Step 5: Substitute values into the Quadratic Formula

Using the values calculated and substituting them into the quadratic formula:\[ r = \frac{-(-8) \pm 14}{2 \cdot 1} \]This simplifies to:\[ r = \frac{8 \pm 14}{2} \]

Step 6: Solve for the values of r

Solve for the two possible values of \(r\):\[ r = \frac{8 + 14}{2} = \frac{22}{2} = 11 \]\[ r = \frac{8 - 14}{2} = \frac{-6}{2} = -3 \]So the solutions to the equation are \(r = 11\) and \(r = -3\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation. It has the general form \( ax^2 + bx + c = 0 \), where:

• \( a \) is the coefficient of \( x^2 \) (quadratic term)
• \( b \) is the coefficient of \( x \) (linear term)
• \( c \) is the constant term.

In our example, \( r^2 - 8r - 33 = 0 \), we can see the coefficients are \( a = 1 \), \( b = -8 \), and \( c = -33 \). The equation has a squared term (\( r^2 \)), making it quadratic. Identifying these coefficients is the first crucial step in solving the equation using the Quadratic Formula.

Discriminant

The discriminant is part of the Quadratic Formula and helps determine the nature of the roots of the quadratic equation. It is given by \( b^2 - 4ac \). In our equation, \( r^2 - 8r - 33 = 0 \), the values are:

• \( a = 1 \)
• \( b = -8 \)
• \( c = -33 \)

When we substitute these into the discriminant formula, we get: \( (-8)^2 - 4(1)(-33) = 64 + 132 = 196 \). The discriminant is 196. Since it is positive, the equation has two distinct real roots. If the discriminant was zero, the equation would have one real root. If it was negative, the roots would be complex.

Solving Equations

Solving quadratic equations can be done using multiple methods. One such method is the Quadratic Formula, which is: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This formula provides a reliable way to find the roots of any quadratic equation.
Using our equation \( r^2 - 8r - 33 = 0 \), we substitute \( a = 1 \), \( b = -8 \), and \( c = -33 \) into the formula, leading to:
\( r = \frac{-(-8) \pm \sqrt{196}}{2(1)} \)
\( r = \frac{8 \pm 14}{2} \)
Solving this gives us two possible values for \( r \):

• \( r = \frac{8 + 14}{2} = 11 \)
• \( r = \frac{8 - 14}{2} = -3 \)

Thus, the solutions are \( r = 11 \) and \( r = -3 \).

Algebra

Algebra involves working with symbols and letters to represent numbers and quantities in formulas and equations. It’s a fundamental part of mathematics that deals with various forms of equations, including quadratic equations.
In solving the quadratic equation \( r^2 - 8r - 33 = 0 \), we used algebraic principles to identify coefficients, compute the discriminant, and apply the quadratic formula to find the roots.

• Identify coefficients \( a, b, c \)
• Calculate the discriminant \( b^2 - 4ac \)
• Substitute into the quadratic formula to find the roots

These steps form a structured approach to solving quadratic equations, demonstrating the power of algebra in solving mathematical problems.