Solve for x in the equation x²+20x + 100 = 36.
Answer (2)
Recognize the perfect square trinomial: ( x + 10 ) 2 = 36 .
Take the square root of both sides: x + 10 = ± 6 .
Solve for x in both cases: x = − 10 + 6 and x = − 10 − 6 .
The solutions are x = − 4 and x = − 16 , so x = − 16 , − 4 .
Explanation
Understanding the Problem We are given the equation x 2 + 20 x + 100 = 36 and asked to solve for x . This looks like a quadratic equation, and we can solve it by completing the square.
Rewriting the Equation Notice that the left side of the equation is a perfect square trinomial. We can rewrite the equation as ( x + 10 ) 2 = 36 .
Taking the Square Root Now, we take the square root of both sides of the equation: ( x + 10 ) 2 = ± 36 . This simplifies to x + 10 = ± 6 .
Two Cases We now have two separate equations to solve for x :
x + 10 = 6
x + 10 = − 6
Solving for x Solving the first equation, we subtract 10 from both sides: x = 6 − 10 = − 4 .
Solving the second equation, we subtract 10 from both sides: x = − 6 − 10 = − 16 .
Final Answer Therefore, the solutions for x are x = − 4 and x = − 16 .
Examples
Completing the square is a useful technique in many areas, such as finding the vertex of a parabola or solving optimization problems. For example, suppose you want to build a rectangular garden with a fixed perimeter. Completing the square can help you find the dimensions that maximize the area of the garden. This technique is also used in physics to analyze oscillatory motion and in engineering to design stable systems.
The solutions to the equation x 2 + 20 x + 100 = 36 are x = − 4 and x = − 16 . This is found by recognizing the left side as a perfect square and taking the square root of both sides. Two cases arise, leading to the final solutions.
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