Use the quadratic formula to find both solutions to the quadratic equation given below.
[tex]4 x^2+5 x+1=0[/tex]
A. [tex]x=\frac{-3-\sqrt{7}}{8}[/tex]
B. [tex]x=\frac{-3-\sqrt{-7}}{8}[/tex]
C. [tex]x=\frac{-3+\sqrt{7}}{8}[/tex]
D. [tex]x=\frac{-3+\sqrt{-7}}{8}[/tex]
E. [tex]x=-\frac{1}{4}[/tex]
F. [tex]x=-1[/tex]
Answer (1)
Identify the coefficients: a = 4 , b = 5 , c = 1 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 2 ( 4 ) − 5 ± 5 2 − 4 ( 4 ) ( 1 ) = 8 − 5 ± 9 = 8 − 5 ± 3 .
Solve for both values of x : x 1 = − 4 1 and x 2 = − 1 . The solutions are − 4 1 , − 1 .
Explanation
Problem Introduction We are given the quadratic equation 4 x 2 + 5 x + 1 = 0 and asked to find its solutions using the quadratic formula.
Quadratic Formula The quadratic formula is given by: x = 2 a − b ± b 2 − 4 a c where a , b , and c are the coefficients of the quadratic equation a x 2 + b x + c = 0 .
Identifying Coefficients In our equation, 4 x 2 + 5 x + 1 = 0 , we identify the coefficients as a = 4 , b = 5 , and c = 1 .
Substituting Values Now, we substitute these values into the quadratic formula: x = 2 ( 4 ) − 5 ± 5 2 − 4 ( 4 ) ( 1 )
Simplifying the Discriminant Simplify the expression under the square root: 5 2 − 4 ( 4 ) ( 1 ) = 25 − 16 = 9
Substituting Back Substitute this value back into the quadratic formula: x = 8 − 5 ± 9
Simplifying the Square Root Since 9 = 3 , we have: x = 8 − 5 ± 3
Finding the Solutions Now, we find the two possible values for x : x 1 = 8 − 5 + 3 = 8 − 2 = − 4 1 x 2 = 8 − 5 − 3 = 8 − 8 = − 1
Final Answer Thus, the solutions to the quadratic equation 4 x 2 + 5 x + 1 = 0 are x = − 4 1 and x = − 1 .
Examples
Quadratic equations are useful in various real-world applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its perimeter and area, or modeling the growth of a population. For example, if you want to build a rectangular garden with a specific area and have a limited amount of fencing, you can use a quadratic equation to find the dimensions of the garden that maximize the area within the constraint of the available fencing. Understanding how to solve quadratic equations allows you to optimize designs and predict outcomes in many practical situations.
