Of course, I'd be happy to help you with the problem! Let's go through the steps to solve the equation -2x^2 + 4x = 9.
Step 1: Start by rearranging the equation to bring all the terms to one side, so we have a quadratic equation in standard form: -2x^2 + 4x - 9 = 0.
Step 2: To solve the quadratic equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / (2a). In this case, a = -2, b = 4, and c = -9.
Step 3: Substitute the values of a, b, and c into the quadratic formula and simplify:
x = (-4 ± √(4^2 - 4(-2)(-9))) / (2(-2))
x = (-4 ± √(16 - 72)) / (-4)
x = (-4 ± √(-56)) / (-4)
Step 4: Simplify the square root of -56. Since the square root of a negative number is not a real number, we can express it as i√56:
x = (-4 ± i√56) / (-4)
Step 5: Simplify the expression by dividing both the numerator and denominator by -4:
x = (4/4) ± (i√56/4)
x = 1 ± (i√14)
So, the solutions to the equation -2x^2 + 4x = 9 are (2 - i√14)/2 and (2 + i√14)/2.
I hope this step-by-step explanation helps! Let me know if you have any other questions.