Solve the equation by using the quadratic formula.
[tex]25 x^2+60 x=-36[/tex]
a. [tex]x=\frac{5}{6}[/tex]
b. [tex]x=\frac{6}{5}[/tex]
c. [tex]x=-\frac{6}{5}[/tex]
d. [tex]x=-\frac{5}{6}[/tex]
Answer (1)
Rewrite the equation in standard form: 25 x 2 + 60 x + 36 = 0 .
Identify coefficients: a = 25 , b = 60 , c = 36 .
Apply the quadratic formula: x = 2 ( 25 ) − 60 ± 6 0 2 − 4 ( 25 ) ( 36 ) .
Simplify to find the root: x = − 5 6 .
The final answer is − 5 6 .
Explanation
Problem Analysis We are given the quadratic equation 25 x 2 + 60 x = − 36 . Our goal is to solve this equation using the quadratic formula and select the correct answer from the given options.
Rewrite the Equation First, we need to rewrite the equation in the standard quadratic form a x 2 + b x + c = 0 . Adding 36 to both sides of the equation, we get: 25 x 2 + 60 x + 36 = 0
Identify Coefficients Now, we identify the coefficients a , b , and c . In this case, a = 25 , b = 60 , and c = 36 .
Apply Quadratic Formula Next, we apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c
Substituting the values of a , b , and c , we have: x = 2 × 25 − 60 ± 6 0 2 − 4 × 25 × 36
Simplify the Expression Now, we simplify the expression: x = 50 − 60 ± 3600 − 3600
x = 50 − 60 ± 0
x = 50 − 60
x = − 5 6
Compare with Options Since the discriminant is zero, there is only one real root, which is x = − 5 6 . Comparing this with the given options, we see that option c matches our solution.
Final Answer Therefore, the solution to the equation 25 x 2 + 60 x = − 36 is x = − 5 6 .
Examples
Quadratic equations are used in various real-life scenarios, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area with a specific perimeter and area, or modeling the growth of a population. For example, if you want to build a rectangular garden with an area of 100 square meters and a perimeter of 40 meters, you can use a quadratic equation to find the length and width of the garden. Understanding how to solve quadratic equations is essential for solving these types of practical problems.
