Find the possible value or values of $r$ in the quadratic equation $r^2-7 r-8=0$.
A) $r=2 / 3, r=5$
B) $r=8, r=-1$
C) $r=-10, r=3$
D) $r=\frac{17+\sqrt{277}}{a}, r=\frac{17-\sqrt{277}}{a}$
Answer (1)
Factor the quadratic equation r 2 − 7 r − 8 = 0 into ( r − 8 ) ( r + 1 ) = 0 .
Set each factor equal to zero: r − 8 = 0 or r + 1 = 0 .
Solve for r : r = 8 or r = − 1 .
The possible values of r are r = 8 , r = − 1 .
Explanation
Understanding the Problem We are given the quadratic equation r 2 − 7 r − 8 = 0 . Our goal is to find the possible values of r that satisfy this equation. We can solve this quadratic equation by factoring.
Factoring the Quadratic Equation To solve the quadratic equation by factoring, we need to find two numbers that multiply to − 8 and add up to − 7 . These numbers are − 8 and 1 . Therefore, we can rewrite the quadratic equation as ( r − 8 ) ( r + 1 ) = 0 .
Finding the Values of r Now, we set each factor equal to zero and solve for r :
r − 8 = 0 or r + 1 = 0
Solving these equations gives us:
r = 8 or r = − 1
Selecting the Correct Option Therefore, the possible values of r are 8 and − 1 . Comparing these values with the given options, we see that option B matches our solution.
Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a ball, determining the dimensions of a rectangular area given its area and a relationship between its sides, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 100 square meters and the length must be 5 meters longer than the width, you can use a quadratic equation to find the dimensions of the garden.
