What is the intermediate step in the form (x+a)^2=b as a result of completing the square for the following equation?
4x ^2 +47x+104=7x
First, let's simplify the equation to make it easier to work with:
4x^2 + 47x + 104 - 7x = 0
This can be simplified to:
4x^2 + 40x + 104 = 0
The intermediate step in completing the square for this equation is to divide the entire equation by the coefficient of x^2, which is 4 in this case:
x^2 + 10x + 26 = 0
Now we can complete the square by adding the square of half of the coefficient of x, which is 5 in this case. Add (5^2) = 25 to both sides of the equation:
x^2 + 10x + 25 + 26 = 25
x^2 + 10x + 25 = -26
Next, we can rewrite the left side of the equation as a perfect square:
(x + 5)^2 = -26
This is the intermediate step after completing the square for the given equation.
To complete the square for the equation 4x^2 + 47x + 104 = 7x, we can follow these steps:
1. Move all the terms to one side of the equation to set it equal to zero:
4x^2 + 47x + 104 - 7x = 0
2. Combine like terms:
4x^2 + 40x + 104 = 0
3. Divide the entire equation by the coefficient of x^2 to make the leading coefficient 1 (optional, but simplifies the process):
x^2 + 10x + 26 = 0
4. Take half of the coefficient of x (10 in this case), square it, and then add and subtract the result inside the equation:
x^2 + 10x + (10/2)^2 - (10/2)^2 + 26 = 0
5. Simplify the equation:
x^2 + 10x + 25 - 25 + 26 = 0
6. Factor the trinomial (x^2 + 10x + 25) and simplify:
(x + 5)^2 + 1 = 0
The intermediate step in the process of completing the square for the equation 4x^2 + 47x + 104 = 7x is (x^2 + 10x + 25) after adding and subtracting (10/2)^2 inside the equation.
To complete the square for the equation 4x^2 + 47x + 104 = 7x, we can follow these steps:
1. Move all terms to one side to get the equation in the form ax^2 + bx + c = 0:
4x^2 + 47x + 104 - 7x = 0
4x^2 + 40x + 104 = 0
2. Divide the entire equation by the coefficient of the squared term, which is 4. This step ensures that the coefficient of the squared term is 1:
(4x^2 + 40x + 104)/4 = 0
x^2 + 10x + 26 = 0
3. Now we can complete the square. Take half of the coefficient of the linear term, which is 10, and square it: (10/2)^2 = 5^2 = 25.
Add 25 to both sides of the equation to maintain the balance:
x^2 + 10x + 25 + 26 = 25
x^2 + 10x + 25 + 26 - 25 = 0
x^2 + 10x + 51 = 0
The intermediate result of completing the square is x^2 + 10x + 51 = 0.