how would you solve
14 = 0.9x(x - 6)
To solve the equation 14 = 0.9x(x - 6), we can follow these steps:
Step 1: Distribute the 0.9 to simplify the equation:
14 = 0.9x^2 - 5.4x
Step 2: Move all terms to one side of the equation:
0.9x^2 - 5.4x - 14 = 0
Step 3: Multiply the entire equation by 10 to eliminate the decimal:
9x^2 - 54x - 140 = 0
Step 4: Now we have a quadratic equation in standard form. There are a few methods to solve this, such as factoring, completing the square, or using the quadratic formula. In this case, let's use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Using the formula, we can determine the values of a, b, and c from the quadratic equation:
a = 9
b = -54
c = -140
Step 5: Plug the values of a, b, and c into the quadratic formula and simplify:
x = (-(-54) ± √((-54)^2 - 4 * 9 * (-140))) / (2 * 9)
x = (54 ± √(2916 + 5040)) / 18
x = (54 ± √(7956)) / 18
x = (54 ± 89.18) / 18
Step 6: Solve for x by calculating both possible solutions:
x1 = (54 + 89.18) / 18
x1 = 6.57
x2 = (54 - 89.18) / 18
x2 = -1.97
So, the solutions to the equation 14 = 0.9x(x - 6) are x = 6.57 and x = -1.97.