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Magazine Chapter 10: Problem 105
Solve by using the Quadratic Formula. \(r^{2}-8 r-33=0\)
Short Answer
Expert verified
The solutions are \(r = 11\) and \(r = -3\).
Step by step solution
01
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 \(c = -33\).
02
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.
03
Calculate the discriminant
First, calculate the discriminant \(b^2 - 4ac\):\[ (-8)^2 - 4(1)(-33) = 64 + 132 = 196 \]The discriminant is 196.
04
Compute the square root of the discriminant
Find the square root of the discriminant 196:\[ \sqrt{196} = 14 \]
05
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:\[ r = \frac{8 \pm 14}{2} \]
06
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 \(r = -3\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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:
- \( a = 1 \)
- \( b = -8 \)
- \( c = -33 \)
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 \( r \):
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 \( r \):
- \( r = \frac{8 + 14}{2} = 11 \)
- \( r = \frac{8 - 14}{2} = -3 \)
Algebra
Algebra involves working with symbols and letters to represent numbers and quantities in formulas and equations. It鈥檚 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.
These steps form a structured approach to solving quadratic equations, demonstrating the power of algebra in solving mathematical problems.
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.
