What are the solution(s) of $-x^2+2 x+3=x^2-2 x+3$?
Answer (1)
Rearrange the equation: − x 2 + 2 x + 3 = x 2 − 2 x + 3 becomes − 2 x 2 + 4 x = 0 .
Factor out the common factor: − 2 x ( x − 2 ) = 0 .
Set each factor to zero: − 2 x = 0 or x − 2 = 0 .
Solve for x : The solutions are x = 0 and x = 2 . Therefore, the solutions are 0 and 2 .
Explanation
Problem Analysis We are given the equation − x 2 + 2 x + 3 = x 2 − 2 x + 3 and asked to find its solutions.
Rearranging the Equation First, let's rearrange the equation to have all terms on one side. We subtract x 2 − 2 x + 3 from both sides: − x 2 + 2 x + 3 − ( x 2 − 2 x + 3 ) = 0
Simplifying Now, simplify the equation: − x 2 + 2 x + 3 − x 2 + 2 x − 3 = 0
− 2 x 2 + 4 x = 0
Factoring Factor out the common factor, which is − 2 x :
− 2 x ( x − 2 ) = 0
Solving for x Set each factor to zero and solve for x :
− 2 x = 0 or x − 2 = 0
Finding the Solutions Solving − 2 x = 0 for x , we get x = 0 .
Solving x − 2 = 0 for x , we get x = 2 .
Final Answer Therefore, the solutions are x = 0 and x = 2 .
Examples
Imagine you are designing a symmetrical bridge where the height on one side is described by − x 2 + 2 x + 3 and the height on the other side is described by x 2 − 2 x + 3 . Finding the solutions to the equation − x 2 + 2 x + 3 = x 2 − 2 x + 3 tells you at which points along the bridge the heights are equal on both sides. This is crucial for ensuring the bridge is properly aligned and balanced. Solving quadratic equations helps engineers ensure structural integrity and symmetry in designs.
