Question

Find the solution(s) to each quadratic equation using completing the square.

Justify your answer by showing and/or explaining your step-by-step work.
1. x² +29 = 10x
2. 3x²
3. 4x² + 19 = 4x= x + 2

Answer 1

To solve the quadratic equation x² + 29 = 10x by completing the square, we rearrange the equation, form a perfect square trinomial, and solve for x, yielding solutions x = 5 + 2i and x = 5 - 2i.

Step-by-step explanation:

The student is asking how to solve quadratic equations by completing the square. This method involves rearranging the equation into a perfect square trinomial and then solving for the variable. Let's take the first quadratic equation x² + 29 = 10x as an example:

1. 
   
3. First, move the linear term to the other side to get x² - 10x + 29 = 0.
4. 
   
6. Next, find the number that makes x² - 10x into a perfect square. This number is (10/2)² = 25.
7. 
   
9. Add and subtract this number inside the equation: x² - 10x + 25 - 25 + 29 = 0.
10. 
    
12. Simplify to get (x - 5)² = -4.
13. 
    
15. Take the square root of both sides to solve for x, resulting in x - 5 = ± 2i (since -4 is a negative number and thus, we introduce the imaginary unit i).
16. 
    
18. Finally, add 5 to both sides of the equation to get x = 5 ± 2i.

The solutions for the equation x² + 29 = 10x are x = 5 + 2i and x = 5 - 2i.