%5E2%3D4%2F9.jpeg "(x-1/4)^2=4/9")
(x - 1/4)^2 = 4/9: A Quadratic Equation Solution
In this article, we will solve the quadratic equation (x - 1/4)^2 = 4/9. This equation involves a square of a binomial expression, which requires careful expansion and simplification to obtain the solution.
Expanding the Binomial Expression
Let's start by expanding the binomial expression (x - 1/4)^2. Using the formula for the square of a binomial, we get:
(x - 1/4)^2 = x^2 - 2(x)(1/4) + (1/4)^2
= x^2 - x/2 + 1/16
Simplifying the Equation
Now, we can set up the equation by equating the expanded expression to 4/9:
x^2 - x/2 + 1/16 = 4/9
To simplify this equation, we can start by multiplying both sides by 16 to eliminate the fractions:
16x^2 - 8x + 1 = 64/9
Multiplying both sides by 9 to eliminate the fraction on the right-hand side, we get:
144x^2 - 72x + 9 = 64
Solving the Quadratic Equation
Now, we can solve this quadratic equation using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 144, b = -72, and c = 9 - 64 = -55. Plugging these values into the quadratic formula, we get:
x = (72 ± √((-72)^2 - 4(144)(-55))) / (2(144))
Simplifying the expression, we get two possible values for x:
x = (72 ± √(5184 + 31680)) / 288
x = (72 ± √36864) / 288
x = (72 ± 192) / 288
x = 1/4 or x = 1 3/4
Therefore, the solutions to the equation (x - 1/4)^2 = 4/9 are x = 1/4 and x = 1 3/4.
