Solve the quadratic equation.
[tex]x^2+9 x-15=0[/tex]

Identify the coefficients: a = 1 , b = 9 , and c = − 15 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ .
Substitute the values and simplify: x = 2 ( 1 ) − 9 ± 9 2 − 4 ( 1 ) ( − 15 ) ​ ​ = 2 − 9 ± 141 ​ ​ .
The solutions are: x = 2 − 9 ± 141 ​ ​ ​ .

Explanation

Problem Analysis We are given the quadratic equation x 2 + 9 x − 15 = 0 . Our goal is to find the values of x that satisfy this equation.

Applying the Quadratic Formula To solve this quadratic equation, we will use the quadratic formula, which 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 . In our case, a = 1 , b = 9 , and c = − 15 .

Substituting the Values Now, we substitute the values of a , b , and c into the quadratic formula: x = 2 ( 1 ) − 9 ± 9 2 − 4 ( 1 ) ( − 15 ) ​ ​

Simplifying the Discriminant Next, we simplify the expression under the square root: 9 2 − 4 ( 1 ) ( − 15 ) = 81 + 60 = 141

Finding the Solutions So, the solutions for x are: x = 2 − 9 ± 141 ​ ​


Examples
Quadratic equations are incredibly useful in various real-world scenarios. For instance, they can model the trajectory of a ball thrown in the air, helping to predict its landing point. They are also used in engineering to design bridges and arches, ensuring structural stability. Furthermore, quadratic equations play a crucial role in economics, where they can model cost and revenue curves to optimize business decisions.

Answered by GinnyAnswer | 2025-07-08