Use the quadratic formula to solve the equation $x^2+6 x+25=0$. Fully simplify your answer, including any non-real solutions.
$x=\square$
Answer (1)
Identify the coefficients: a = 1 , b = 6 , c = 25 .
Apply the quadratic formula: x = 2 a − b ± b 2 − 4 a c .
Substitute and simplify: x = 2 ( 1 ) − 6 ± 6 2 − 4 ( 1 ) ( 25 ) = 2 − 6 ± − 64 .
Simplify the complex roots: x = 2 − 6 ± 8 i = − 3 ± 4 i . The solutions are − 3 + 4 i , − 3 − 4 i .
Explanation
Identifying Coefficients and the Quadratic Formula We are given the quadratic equation x 2 + 6 x + 25 = 0 . Our goal is to solve for x using the quadratic formula. The quadratic formula is a general method for finding the roots (solutions) of any quadratic equation of the form a x 2 + b x + c = 0 , and it is given by:
x = 2 a − b ± b 2 − 4 a c
In our equation, we can identify the coefficients as a = 1 , b = 6 , and c = 25 .
Substituting into the Formula Now, we substitute the values of a , b , and c into the quadratic formula:
x = 2 ( 1 ) − 6 ± 6 2 − 4 ( 1 ) ( 25 )
Let's simplify the expression step by step.
Simplifying the Discriminant First, we simplify the expression under the square root:
6 2 − 4 ( 1 ) ( 25 ) = 36 − 100 = − 64
So, we have:
x = 2 − 6 ± − 64
Dealing with Complex Numbers Since we have a negative number under the square root, we will have complex solutions. Recall that − 1 = i , so we can simplify − 64 as follows:
− 64 = 64 ⋅ − 1 = 8 i
Now, substitute this back into the equation:
x = 2 − 6 ± 8 i
Final Solutions Finally, we divide both the real and imaginary parts by 2:
x = 2 − 6 ± 2 8 i = − 3 ± 4 i
Thus, the solutions are x = − 3 + 4 i and x = − 3 − 4 i .
Conclusion Therefore, the solutions to the quadratic equation x 2 + 6 x + 25 = 0 are x = − 3 + 4 i and x = − 3 − 4 i .
Examples
Quadratic equations are not just abstract math; they appear in various real-world applications. For instance, when designing a parabolic reflector for a flashlight or satellite dish, engineers use quadratic equations to determine the optimal shape and focus. Similarly, in physics, projectile motion, like the trajectory of a ball thrown in the air, can be modeled using quadratic equations. By solving these equations, one can predict the range, maximum height, and time of flight of the projectile. These applications demonstrate how understanding quadratic equations can help solve practical problems in engineering and physics.
