What is the solution to the following equation?
$x^2+5 x+7=0$
A. $x=\frac{-3 \pm \sqrt{-7}}{2}$
B. $x=\frac{3 \pm \sqrt{25}}{2}$
C. $x=\frac{-5 \pm \sqrt{53}}{2}$
D. $x=\frac{-5 \pm \sqrt{-3}}{2}$

Identify the coefficients: a = 1 , b = 5 , c = 7 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c ​ ​ .
Substitute the values: x = 2 ( 1 ) − 5 ± 5 2 − 4 ( 1 ) ( 7 ) ​ ​ .
Simplify to find the solution: x = 2 − 5 ± − 3 ​ ​ .

Explanation

Problem Analysis We are given a quadratic equation x 2 + 5 x + 7 = 0 and asked to find its solution. We can use the quadratic formula to solve this equation.

Quadratic Formula The quadratic formula is given by x = 2 a − b ± b 2 − 4 a c ​ ​ for a quadratic equation in the form a x 2 + b x + c = 0 . In our case, we have a = 1 , b = 5 , and c = 7 .

Substitution Substituting the values of a , b , and c into the quadratic formula, we get: x = 2 ( 1 ) − 5 ± 5 2 − 4 ( 1 ) ( 7 ) ​ ​

Simplification Now, we simplify the expression: x = 2 − 5 ± 25 − 28 ​ ​ x = 2 − 5 ± − 3 ​ ​

Final Answer Comparing our solution with the given options, we see that it matches option D.


Examples
Quadratic equations are used in various real-life applications, such as calculating the trajectory of a projectile, determining the dimensions of a rectangular area given its perimeter and area, and modeling the growth or decay of populations. For example, if you want to launch a rocket to hit a target, you need to solve a quadratic equation to find the launch angle. Similarly, quadratic equations are used in physics, engineering, economics, and computer science to model and solve a wide range of problems.

Answered by GinnyAnswer | 2025-07-08