Select the correct answer.
What are the solutions of this quadratic equation?
[tex]4 x^2+3=4 x+2[/tex]
A. [tex]x=\frac{1}{2}[/tex]
B. [tex]x=-2 \pm \sqrt{3}[/tex]
C. [tex]x=2[/tex]
D. [tex]x=\frac{1 \pm \sqrt{3}}{2}[/tex]
Answer (2)
Rewrite the equation in standard form: 4 x 2 − 4 x + 1 = 0 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute the values a = 4 , b = − 4 , and c = 1 into the formula.
Simplify to find the solution: x = 2 1 .
x = 2 1
Explanation
Problem Analysis We are given the quadratic equation 4 x 2 + 3 = 4 x + 2 . Our goal is to find the solutions for x .
Rewriting the Equation First, we need to rewrite the equation in the standard quadratic form, which is a x 2 + b x + c = 0 . To do this, we subtract 4 x and 2 from both sides of the equation:
4 x 2 + 3 − 4 x − 2 = 0
4 x 2 − 4 x + 1 = 0
Applying the Quadratic Formula Now we have the equation in the standard form, with a = 4 , b = − 4 , and c = 1 . We can use the quadratic formula to find the solutions for x . The quadratic formula is:
x = 2 a − b ± b 2 − 4 a c
Calculating the Solutions Substitute the values of a , b , and c into the quadratic formula:
x = 2 ( 4 ) − ( − 4 ) ± ( − 4 ) 2 − 4 ( 4 ) ( 1 )
x = 8 4 ± 16 − 16
x = 8 4 ± 0
x = 8 4 ± 0
x = 8 4
x = 2 1
Final Solution Since the discriminant (the value inside the square root) is 0, we have only one real solution for x , which is x = 2 1 .
Examples
Quadratic equations are used in various real-life situations, such as calculating the trajectory of a ball, determining the dimensions of a garden to maximize area, or modeling the path of a projectile. Understanding how to solve quadratic equations allows us to make accurate predictions and solve optimization problems in these scenarios. For example, if you want to build a rectangular garden with a fixed perimeter, you can use a quadratic equation to find the dimensions that give you the largest possible area.
The solutions of the quadratic equation 4 x 2 + 3 = 4 x + 2 can be found by rewriting it in standard form and applying the quadratic formula. After completing the calculations, the solution is x = 2 1 , which corresponds to option A. Therefore, the answer is x = 2 1 .
;
