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
The intermediate step is to move the constant term to the other side of the equation.
Rewriting the equation:
4x^2 + 47x + 104 - 7x = 0
Simplifying:
4x^2 + 40x + 104 = 0
To complete the square for the equation 4x^2 + 47x + 104 = 7x, we can follow these steps:
1. Move all 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 + (47x - 7x) + 104 = 0
4x^2 + 40x + 104 = 0
3. Divide the entire equation by the coefficient of the x^2 term (in this case, 4) to simplify the equation:
x^2 + (40/4)x + 104/4 = 0
x^2 + 10x + 26 = 0
At this point, the equation is in the form x^2 + bx + c = 0, which is ready for completing the square.
To find the intermediate step in completing the square for the given equation, we first need to rearrange the equation in the form (x + a)^2 = b. Here's how to do it step by step:
1. Start with the given equation: 4x^2 + 47x + 104 = 7x.
2. Move all terms to one side of the equation to make it equal to zero: 4x^2 + 47x + 104 - 7x = 0.
3. Combine like terms: 4x^2 + 40x + 104 = 0.
4. Divide the entire equation by the coefficient of x^2 to make the leading coefficient 1: x^2 + (40/4)x + (104/4) = 0.
5. Simplify: x^2 + 10x + 26 = 0.
Now, we can identify the intermediate step in completing the square. It involves taking half of the coefficient of x, squaring it, and adding/subtracting it to both sides of the equation.
6. Take half of the coefficient of x, which is 10. Square it to get 100.
7. Add 100 to both sides of the equation: x^2 + 10x + 100 + 26 = 0 + 100.
8. Simplify: x^2 + 10x + 126 = 100.
At this step (step 8), we have completed the square for the equation. The intermediate step would be adding 100 to both sides of the equation to create a perfect square trinomial on the left side.