Which graph can be used to find the solution(s) to [tex]$x^2-1=3$[/tex]?
Answer (2)
Rewrite the equation as x 2 − 4 = 0 and graph f ( x ) = x 2 − 4 to find the x-intercepts.
Alternatively, rewrite the equation as x 2 = 4 and graph f ( x ) = x 2 and g ( x ) = 4 to find the intersection points.
Both graphs can be used to find the solutions to the equation x 2 − 1 = 3 .
The solutions are − 2 and 2 .
Explanation
Understanding the Problem We are asked to find which graph can be used to solve the equation x 2 − 1 = 3 . There are a couple of approaches we can take to solve this problem graphically.
Method 1: Finding x-intercepts One approach is to rewrite the equation as x 2 − 4 = 0 . Then, we can consider the function f ( x ) = x 2 − 4 . The solutions to the equation x 2 − 4 = 0 are the x-intercepts of the graph of f ( x ) .
Method 2: Finding Intersection Points Another approach is to rewrite the equation as x 2 = 4 . Then, we can consider two functions: f ( x ) = x 2 and g ( x ) = 4 . The solutions to the equation x 2 = 4 are the x-coordinates of the points where the graphs of f ( x ) and g ( x ) intersect.
Conclusion Therefore, we can either graph f ( x ) = x 2 − 4 and find its x-intercepts, or graph f ( x ) = x 2 and g ( x ) = 4 and find the x-coordinates of their intersection points. Both graphs can be used to find the solution(s) to x 2 − 1 = 3 . The solutions are x = − 2 and x = 2 .
Examples
Imagine you are designing a square garden and want to know the side length needed to cover a specific area. If the area can be modeled by the equation x 2 − 1 = 3 , where x is related to the side length, solving this equation helps you determine the exact dimensions for your garden. Graphing the related functions visually confirms the possible side lengths that satisfy the area requirement, providing a practical application of quadratic equations in real-world design scenarios.
To find the solutions to the equation x 2 − 1 = 3 , you can graph the function f ( x ) = x 2 − 4 and find the x-intercepts, or graph g ( x ) = x 2 and h ( x ) = 4 to find intersection points. The solutions are x = − 2 and x = 2 .
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