Find all real solutions. (Enter your answers as comma-separated lists. If there is no real solution, enter NO REALSOLUTION.)
[tex]x^2-x-30=0[/tex]
Answer (1)
Factor the quadratic equation: x 2 − x − 30 = ( x − 6 ) ( x + 5 ) .
Set each factor to zero: x − 6 = 0 or x + 5 = 0 .
Solve for x : x = 6 or x = − 5 .
The real solutions are − 5 , 6 .
Explanation
Understanding the Problem We are given the quadratic equation x 2 − x − 30 = 0 . Our goal is to find all real solutions for x . We can solve this by factoring the quadratic expression.
Factoring the Quadratic We need to factor the quadratic expression x 2 − x − 30 . We are looking for two numbers that multiply to -30 and add to -1. These numbers are -6 and 5. Therefore, we can write the quadratic expression as ( x − 6 ) ( x + 5 ) .
Solving for x Now we set each factor equal to zero and solve for x :
x − 6 = 0 or x + 5 = 0
Solving these equations, we get:
x = 6 or x = − 5
Final Answer The real solutions for the equation x 2 − x − 30 = 0 are x = 6 and x = − 5 .
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 given its area and perimeter, or modeling growth and decay processes. For example, if you want to build a rectangular garden with an area of 30 square meters and the length should be 1 meter longer than the width, you can use a quadratic equation to find the exact dimensions of the garden. Understanding how to solve quadratic equations helps in optimizing designs and predicting outcomes in many practical situations.
