Question

Use the method of completing the square to transform the quadratic equation into the equation form (x – p)2 = q.

x2 - 12x - 9 = 0
A) (x - 36)2 = 9
B) (x - 36)2 = -9
C) (x - 6)2 = 45
D) (x - 6)2 = -45

Answer 1

x^2-12x-9=0
Add 9 to both sides
x^2-12x=9
Add (b/2)^2 to both sides
x^2-12x+36=45
Factor
(x-6)^2=45
So C is the answer

Answer 2

To complete the square for the quadratic equation x²−12x−9=0, we follow a method that involves adding and subtracting the square of half the coefficient of the linear term (b). The given equation is: x²−12 x−9=0

First, move the constant term to the other side of the equation: x²−12 x=9

Now, to complete the square, add and subtract (12/2)² to the expression: x²−12x+(12/2)²=9+(12/2)²

This simplifies to: (x−6)²=45

So, the transformed equation is (x−6)²=45, which is in the desired form (x−p)²=q. Therefore, the correct answer is: C) (x−6)²=45

This completion of the square technique is a useful method for rewriting a quadratic equation in a perfect square trinomial form, making it easier to analyse and solve. The key is to manipulate the equation by adding and subtracting the appropriate value to create a perfect square on one side.

Completing the square transforms, the quadratic equation x²−12x−9=0 into the equivalent form (x−6)²=45, showcasing the symmetry of the resulting perfect square trinomial. This method facilitates a more convenient approach to solving quadratic equations and provides insights into the graphical representation of the equation on the coordinate plane.