What is the solution set of the equation [tex]$x^2+3 x-4=6 ?$[/tex]
A. [tex]$\{-5,2\}$[/tex]
B. [tex]$\{-2,-1\}$[/tex]
C. [tex]$\{2,7\}$[/tex]
D. [tex]$(5,10)$[/tex]
Answer (1)
Rewrite the equation in standard form: x 2 + 3 x − 10 = 0 .
Factor the quadratic expression: ( x + 5 ) ( x − 2 ) = 0 .
Set each factor to zero and solve for x : x + 5 = 0 or x − 2 = 0 , which gives x = − 5 or x = 2 .
The solution set is { − 5 , 2 } .
Explanation
Understanding the Problem We are given the quadratic equation x 2 + 3 x − 4 = 6 . Our goal is to find the solution set for x . This means we need to find all values of x that satisfy the equation.
Rewriting the Equation First, we need to rewrite the equation in the standard quadratic form, which is a x 2 + b x + c = 0 . To do this, we subtract 6 from both sides of the equation:
x 2 + 3 x − 4 − 6 = 6 − 6
This simplifies to:
x 2 + 3 x − 10 = 0
Factoring the Quadratic Now, we need to factor the quadratic expression x 2 + 3 x − 10 . We are looking for two numbers that multiply to -10 and add to 3. These numbers are 5 and -2, since 5 × − 2 = − 10 and 5 + ( − 2 ) = 3 . Therefore, we can rewrite the quadratic as:
( x + 5 ) ( x − 2 ) = 0
Solving for x To find the solutions for x , we set each factor equal to zero and solve for x :
x + 5 = 0 or x − 2 = 0
Solving for x in each case:
x = − 5 or x = 2
The Solution Set Therefore, the solution set is { − 5 , 2 } . This means that the values x = − 5 and x = 2 satisfy the original equation x 2 + 3 x − 4 = 6 .
Examples
Quadratic equations are incredibly useful in various real-world scenarios. For example, engineers use them to design bridges and calculate the forces acting on them. In business, quadratic equations can help model profit and revenue to optimize pricing strategies. Even in sports, understanding the trajectory of a ball involves solving quadratic equations. By mastering these equations, you're unlocking tools applicable in many fields!
