What are the solutions to $x^2+8 x+7=0$?
A. $x=-8$ and $x=-7$
B. $x=-7$ and $x=-1$
C. $x=1$ and $x=7$
D. $x=7$ and $x=8$
Answer (1)
Factor the quadratic equation x 2 + 8 x + 7 = 0 into ( x + 1 ) ( x + 7 ) = 0 .
Set each factor equal to zero: x + 1 = 0 or x + 7 = 0 .
Solve for x in each case: x = − 1 or x = − 7 .
The solutions to the quadratic equation are x = − 7 and x = − 1 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 8 x + 7 = 0 . Our goal is to find the values of x that satisfy this equation. We can solve this by factoring the quadratic expression.
Factoring the Quadratic We need to factor the quadratic expression x 2 + 8 x + 7 . We are looking for two numbers that multiply to 7 and add to 8. These numbers are 1 and 7. Therefore, we can write the quadratic expression as ( x + 1 ) ( x + 7 ) .
Setting Factors to Zero Now we set the factored expression equal to zero: ( x + 1 ) ( x + 7 ) = 0 . This means that either ( x + 1 ) = 0 or ( x + 7 ) = 0 .
Solving for x Solving for x in each case: If x + 1 = 0 , then x = − 1 .
If x + 7 = 0 , then x = − 7 .
Thus, the solutions are x = − 1 and x = − 7 .
Examples
Quadratic equations are used in many real-world applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its perimeter and area, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 100 square feet and you know one side must be 5 feet longer than the other, you can use a quadratic equation to find the dimensions of the garden.
