Solve the equation using the quadratic formula. [tex]x^2+13 x+7=0[/tex] The solution set is { }. (Simplify your answer. Type an exact answer, using radicals and [tex]i[/tex] as needed. Use integers or fractions for any numbers in the expression.)
Answer (2)
Identify the coefficients: a = 1 , b = 13 , and c = 7 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute the values: x = 2 ( 1 ) − 13 ± 1 3 2 − 4 ( 1 ) ( 7 ) .
Simplify to find the solutions: x = 2 − 13 − 141 and x = 2 − 13 + 141 . The solution set is { 2 − 13 − 141 , 2 − 13 + 141 } .
Explanation
Understanding the Problem and the Quadratic Formula We are given the quadratic equation x 2 + 13 x + 7 = 0 and asked to find its solutions using the quadratic formula. The quadratic formula is a general method for solving quadratic equations of the form a x 2 + b x + c = 0 , where a , b , and c are constants. The formula is given by:
x = 2 a − b ± b 2 − 4 a c
Identifying Coefficients and Substituting into the Formula In our equation, x 2 + 13 x + 7 = 0 , we can identify the coefficients as follows:
a = 1 b = 13 c = 7
Now, we substitute these values into the quadratic formula:
x = 2 ( 1 ) − 13 ± 1 3 2 − 4 ( 1 ) ( 7 )
Simplifying the Expression Next, we simplify the expression inside the square root:
1 3 2 − 4 ( 1 ) ( 7 ) = 169 − 28 = 141
So, we have:
x = 2 − 13 ± 141
Since 141 has no perfect square factors other than 1, 141 cannot be simplified further.
Finding the Solutions Therefore, the two solutions for x are:
x = 2 − 13 + 141 and x = 2 − 13 − 141
The solution set is thus { 2 − 13 + 141 , 2 − 13 − 141 } .
Final Answer The solution set for the quadratic equation x 2 + 13 x + 7 = 0 using the quadratic formula is:
{ 2 − 13 − 141 , 2 − 13 + 141 }
Examples
The quadratic formula is not just an abstract mathematical concept; it has real-world applications. For instance, engineers use it to calculate the trajectory of a projectile, like a ball thrown in the air or a rocket launched into space. By knowing the initial velocity, launch angle, and gravitational forces, they can predict where the projectile will land. Similarly, in finance, the quadratic formula can be used to model investment growth or calculate loan payments, helping individuals and businesses make informed financial decisions. These examples highlight how understanding and applying the quadratic formula can solve practical problems in various fields.
The solutions to the quadratic equation x 2 + 13 x + 7 = 0 using the quadratic formula are x = 2 − 13 ± 141 . The solution set is { 2 − 13 − 141 , 2 − 13 + 141 } .
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