To find the intermediate step in completing the square for the given equation 6x^2 + 46x + 312 = 14x, we should follow these steps:
1. Start with the given equation: 6x^2 + 46x + 312 = 14x.
2. Move all terms to one side of the equation to set up for completing the square. Subtract 14x from both sides: 6x^2 + 46x - 14x + 312 = 0.
3. Combine like terms: 6x^2 + 32x + 312 = 0.
4. Factor out the coefficient of x^2 (6) from the x terms: 6(x^2 + (32/6)x) + 312 = 0.
5. Now, focus on completing the square for the expression inside the parentheses: x^2 + (32/6)x.
6. To complete the square, take half of the coefficient of x (32/6 = 16/3) and square it ((16/3)^2 = 256/9).
7. Add and subtract the squared value inside the parentheses: x^2 + (32/6)x + 256/9 - 256/9.
8. Rewrite the equation with the completed square form: 6[(x + 16/3)^2 - 256/9] + 312 = 0.
This is the intermediate step in completing the square for the given equation 6x^2 + 46x + 312 = 14x. It involves transforming the equation to a perfect square trinomial by adding and subtracting the square of half the coefficient of x.
