Solve the equation by factoring.
[tex]x^2=5 x+14[/tex]
The solution set is { }. (Use a comma to separate answers as needed.)
Answer (2)
Rewrite the equation in standard form: x 2 − 5 x − 14 = 0 .
Factor the quadratic expression: ( x − 7 ) ( x + 2 ) = 0 .
Set each factor to zero and solve for x : x = 7 or x = − 2 .
The solution set is { − 2 , 7 } .
Explanation
Understanding the Problem We are given the quadratic equation x 2 = 5 x + 14 . Our goal is to solve this equation by factoring. This means we want to rewrite the equation in the form ( x − a ) ( x − b ) = 0 , where a and b are the solutions to the equation.
Rewriting the Equation First, we need to rewrite the equation in the standard form of a quadratic equation, which is a x 2 + b x + c = 0 . To do this, we subtract 5 x and 14 from both sides of the equation:
x 2 − 5 x − 14 = 0
Factoring the Quadratic Expression Now, we need to factor the quadratic expression x 2 − 5 x − 14 . We are looking for two numbers that multiply to − 14 and add to − 5 . These numbers are − 7 and 2 . So, we can factor the expression as:
( x − 7 ) ( x + 2 ) = 0
Solving for x Next, we set each factor equal to zero and solve for x :
x − 7 = 0 or x + 2 = 0
Solving these equations, we get:
x = 7 or x = − 2
The Solution Set Therefore, the solution set is { − 2 , 7 } .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area with a specific perimeter and area, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 14 square meters and you know one side must be 5 meters longer than the other, you can use a quadratic equation to find the exact dimensions of the garden. Factoring helps simplify these equations to find practical solutions.
The solutions to the equation x 2 = 5 x + 14 are found by rewriting it in standard form and factoring. This results in the factors ( x − 7 ) ( x + 2 ) = 0 , leading to the solution set {-2, 7}. Thus, the values of x that satisfy the equation are − 2 and 7 .
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