$(x+\square)^2=\square
Answer (1)
Rewrite the incomplete quadratic equation using constants a and b : ( x + a ) 2 = b .
Expand the left side: x 2 + 2 a x + a 2 = b .
Rearrange the terms: x 2 + 2 a x + ( a 2 − b ) = 0 .
Solve for x using the quadratic formula: x = − a ± b .
Explanation
Understanding the Problem We are given an incomplete quadratic equation in the form ( x + □ ) 2 = □ , where the squares represent unknown constants. Our goal is to understand the relationship between these constants and explore possible solutions for x .
Rewriting the Equation Let's represent the unknown constant inside the parenthesis as a and the unknown constant on the right side of the equation as b . The equation then becomes: ( x + a ) 2 = b
Expanding the Square Expanding the left side of the equation, we get: x 2 + 2 a x + a 2 = b
Rearranging Terms Rearranging the equation to isolate the constant terms, we have: x 2 + 2 a x + ( a 2 − b ) = 0
Solving for x Now, we can solve for x using the quadratic formula: x = 2 ( 1 ) − 2 a ± ( 2 a ) 2 − 4 ( 1 ) ( a 2 − b ) Simplifying this, we get: x = 2 − 2 a ± 4 a 2 − 4 a 2 + 4 b x = 2 − 2 a ± 4 b x = 2 − 2 a ± 2 b x = − a ± b
Expressing b in terms of a Thus, the solutions for x are x = − a + b and x = − a − b . Alternatively, we can express b in terms of a . If we let b = a 2 , then the equation becomes ( x + a ) 2 = a 2 . Taking the square root of both sides, we get x + a = ± a , so x = − a + a = 0 or x = − a − a = − 2 a .
Final Answer In conclusion, the solutions for x depend on the values of a and b . We have found that x = − a ± b .
Examples
Understanding how to complete the square and solve quadratic equations is crucial in many fields, such as physics and engineering. For instance, when analyzing projectile motion, you often need to solve quadratic equations to determine the time it takes for an object to reach a certain height. By manipulating the equation into a completed square form, you can easily find the vertex of the parabola, which represents the maximum height of the projectile. This skill is also useful in optimizing various systems, such as electrical circuits or mechanical designs, where quadratic relationships are common.
