Which is the solution of the quadratic equation $(4 y-3)^2=72$?
A. $y=\frac{3+5 \sqrt{2}}{4}$ and $y=\frac{3-5 \sqrt{2}}{4}$
B. $y=\frac{3+6 \sqrt{2}}{4}$ and $y=\frac{-3-6 \sqrt{2}}{4}$
C. $y=\frac{9 \sqrt{2}}{4}$ and $y=\frac{-3 \sqrt{2}}{4}$
D. $y=\frac{9 \sqrt{2}}{4}$ and $y=\frac{3 \sqrt{2}}{4}$
Answer (1)
Take the square root of both sides of the equation: 4 y − 3 = ± 72 .
Simplify the square root: 4 y − 3 = ± 6 2 .
Solve for y : y = 4 3 ± 6 2 .
The solutions are y = 4 3 + 6 2 and y = 4 3 − 6 2 .
Explanation
Problem Analysis We are given the quadratic equation ( 4 y − 3 ) 2 = 72 and asked to find its solutions.
Taking Square Root To solve this equation, we first take the square root of both sides: ( 4 y − 3 ) 2 = ± 72
Simplifying the Radical Simplifying the square root of 72, we have 72 = 36 × 2 = 6 2 . Thus, the equation becomes: 4 y − 3 = ± 6 2
Isolating y Now, we solve for y by adding 3 to both sides and then dividing by 4: 4 y = 3 ± 6 2 y = 4 3 ± 6 2
Final Solutions Therefore, the two solutions for y are: y = 4 3 + 6 2 and y = 4 3 − 6 2
Examples
Quadratic equations like this one appear in various contexts, such as determining the dimensions of a rectangle with a given area or modeling projectile motion. For example, if the area of a square is represented by ( 4 y − 3 ) 2 and we know the area is 72 square units, we can solve this equation to find the value of y , which could represent a side length or another relevant parameter. Understanding how to solve quadratic equations is crucial in many fields, including engineering, physics, and economics.
