Solve $x^2+\frac{1}{2} x+\frac{1}{16}=\frac{4}{9}$.
Factor the perfect-square trinomial on the left side of the equation.
$(x+\square)^2=\frac{4}{9}$
Answer (1)
Factor the perfect square trinomial: x 2 + 2 1 x + 16 1 = ( x + 4 1 ) 2 .
Rewrite the equation: ( x + 4 1 ) 2 = 9 4 .
Take the square root of both sides: x + 4 1 = ± 3 2 .
Solve for x: x = 12 5 and x = − 12 11 .
x = 12 5 , x = − 12 11
Explanation
Problem Analysis We are given the equation x 2 + 2 1 x + 16 1 = 9 4 . Our goal is to solve for x by factoring the perfect-square trinomial on the left side of the equation.
Factoring the Trinomial First, we need to factor the perfect square trinomial x 2 + 2 1 x + 16 1 . We recognize that this is in the form a 2 + 2 ab + b 2 = ( a + b ) 2 . In this case, a = x and we need to find b such that 2 ab = 2 1 x . So, 2 x b = 2 1 x , which means b = 4 1 . Thus, x 2 + 2 1 x + 16 1 = ( x + 4 1 ) 2 .
Rewriting the Equation Now we can rewrite the equation as ( x + 4 1 ) 2 = 9 4 .
Taking the Square Root Next, we take the square root of both sides of the equation: ( x + 4 1 ) 2 = ± 9 4 . This simplifies to x + 4 1 = ± 3 2 .
Solving for x Now we solve for x in both cases:
Case 1: x + 4 1 = 3 2 . Subtract 4 1 from both sides: x = 3 2 − 4 1 . To combine these fractions, we need a common denominator, which is 12. So, x = 12 8 − 12 3 = 12 5 .
Case 2: x + 4 1 = − 3 2 . Subtract 4 1 from both sides: x = − 3 2 − 4 1 . Again, we need a common denominator, which is 12. So, x = − 12 8 − 12 3 = − 12 11 .
Final Solutions Therefore, the solutions are x = 12 5 and x = − 12 11 .
Examples
Imagine you're designing a square garden and want to expand it while keeping it a perfect square. This problem helps you determine how much to increase the side length of the original garden to reach a specific new area. Understanding quadratic equations and perfect square trinomials is crucial in various fields like engineering, physics, and computer science, where precise calculations are essential for designing structures, modeling physical phenomena, and developing algorithms.
