Which graph can be used to find the solution(s) to $x^2-4 x+4=2 x+1+x^2 $?
Answer (1)
Simplify the equation x 2 − 4 x + 4 = 2 x + 1 + x 2 to − 6 x + 3 = 0 .
Solve for x to find x = 2 1 .
Graph y = x 2 − 4 x + 4 and y = 2 x + 1 + x 2 and find the intersection point.
Alternatively, graph y = − 6 x + 3 and find the x-intercept, which is 2 1 .
Explanation
Understanding the Problem We are given the equation x 2 − 4 x + 4 = 2 x + 1 + x 2 . Our goal is to determine which graph can be used to find the solution(s) to this equation.
Simplifying the Equation First, let's simplify the equation by subtracting x 2 from both sides: x 2 − 4 x + 4 − x 2 = 2 x + 1 + x 2 − x 2 This simplifies to: − 4 x + 4 = 2 x + 1
Isolating x Next, we want to isolate x . Add 4 x to both sides of the equation: − 4 x + 4 + 4 x = 2 x + 1 + 4 x 4 = 6 x + 1
Further Isolating x Subtract 1 from both sides: 4 − 1 = 6 x + 1 − 1 3 = 6 x
Solving for x Finally, divide both sides by 6: 6 3 = 6 6 x x = 2 1
Determining the Graph Now, let's consider the graphs that could be used to find the solution.
We could graph y = x 2 − 4 x + 4 and y = 2 x + 1 + x 2 and find their intersection point. The x-coordinate of the intersection point would be the solution.
We could rearrange the simplified equation − 6 x + 3 = 0 or 6 x − 3 = 0 . Then, we can graph y = − 6 x + 3 or y = 6 x − 3 and find the x-intercept (where y = 0 ).
Since the question asks which graph can be used, any of these options would work. However, the simplest graph to use would be either y = − 6 x + 3 or y = 6 x − 3 , as we only need to find the x-intercept.
Finding the Solution Graphically The solution to the equation is x = 2 1 . We can graph y = x 2 − 4 x + 4 and y = 2 x + 1 + x 2 and find the intersection point. Alternatively, we can graph y = − 6 x + 3 and find the x-intercept. Another alternative is to graph y = 6 x − 3 and find the x-intercept.
Examples
In electrical engineering, you might need to find the point where two circuits have the same voltage output. By setting the voltage equations of the two circuits equal to each other, you create an equation similar to the one we solved. Graphing both voltage equations and finding their intersection point visually identifies the operating point where both circuits behave identically, ensuring proper system integration and function.
