Solve the following by completing the square:
1) [tex]$x^2-2 x-19=0$[/tex]
2) [tex]$x^2-7 x+12=0$[/tex]
Answer (1)
For the equation x 2 − 2 x − 19 = 0 , complete the square to get ( x − 1 ) 2 = 20 , then solve for x to find x = 1 ± 2 5 .
For the equation x 2 − 7 x + 12 = 0 , complete the square to get ( x − 2 7 ) 2 = 4 1 , then solve for x to find x = 3 , 4 .
The solutions for the first equation are x = 1 + 2 5 and x = 1 − 2 5 .
The solutions for the second equation are x = 3 and x = 4 .
x = 1 ± 2 5 , x = 3 , 4
Explanation
Problem Analysis We are given two quadratic equations to solve by completing the square:
x 2 − 2 x − 19 = 0
x 2 − 7 x + 12 = 0
Solving the first equation For the first equation, x 2 − 2 x − 19 = 0 , we want to rewrite it in the form ( x − h ) 2 = k . First, we add 19 to both sides of the equation:
x 2 − 2 x = 19
To complete the square, we need to add ( 2 − 2 ) 2 = ( − 1 ) 2 = 1 to both sides:
x 2 − 2 x + 1 = 19 + 1
This simplifies to:
( x − 1 ) 2 = 20
Now, we take the square root of both sides:
x − 1 = ± 20 = ± 2 5
Finally, we solve for x :
x = 1 ± 2 5
Solving the second equation For the second equation, x 2 − 7 x + 12 = 0 , we follow a similar process. First, subtract 12 from both sides:
x 2 − 7 x = − 12
To complete the square, we need to add ( 2 − 7 ) 2 = 4 49 to both sides:
x 2 − 7 x + 4 49 = − 12 + 4 49
This simplifies to:
( x − 2 7 ) 2 = − 4 48 + 4 49 = 4 1
Now, we take the square root of both sides:
x − 2 7 = ± 4 1 = ± 2 1
Finally, we solve for x :
x = 2 7 ± 2 1
So, x = 2 7 + 1 = 2 8 = 4 or x = 2 7 − 1 = 2 6 = 3
Final Answer The solutions to the first equation are x = 1 + 2 5 and x = 1 − 2 5 . The solutions to the second equation are x = 4 and x = 3 .
Examples
Completing the square is a useful technique in many areas of mathematics. For example, in physics, when analyzing the motion of a projectile under gravity, completing the square can help determine the maximum height reached by the projectile. In engineering, it can be used to optimize the design of parabolic reflectors, such as those used in satellite dishes or solar concentrators. By expressing the quadratic equation in vertex form, engineers can easily find the focal point of the parabola, which is crucial for efficient energy collection or signal transmission. This method provides a clear and structured approach to solving quadratic equations, making it a valuable tool in various practical applications.
