0=-3x^2+18x+27
A) one solution
B) two solutions
C) no solutions
D) infinitely many solutions
To solve the equation 0=-3x^2+18x+27, we can start by factoring out -3 from the equation to get:
0 = -3(x^2-6x-9)
Next, we can use the quadratic formula to find the roots of the expression inside the parentheses:
x = (6 ± √(6^2 - 4(-9)))/2
x = (6 ± √72)/2
x = 3 ± 3√2
Therefore, the equation has two solutions, answer choice (B).
Since it only wants to know about the nature of the solutions, of
-3x^2+18x+27 = 0 or
x^2 - 6x - 9 = 0
b^2 - 4ac
= 36 - 4(1)(-9) > 0
so we have 2 real solutions.
If a and c have opposite signs, the discriminant is positive -- so 2 real roots
To determine the number of solutions to the quadratic equation 0 = -3x^2 + 18x + 27, we can use the discriminant. The discriminant is the value inside the square root of the quadratic formula, which can be calculated as b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.
In this case, the equation is in the form ax^2 + bx + c = 0. Taking the coefficients from the given equation, we have a = -3, b = 18, and c = 27. Now, let's calculate the discriminant:
Discriminant (D) = b^2 - 4ac
= (18)^2 - 4(-3)(27)
= 324 + 324
= 648
The value of the discriminant is 648 in this case.
Now, based on the value of the discriminant, we can determine the number of solutions:
1. If the discriminant is larger than 0 (D > 0), it means that the equation has two distinct solutions.
2. If the discriminant is equal to 0 (D = 0), it means that the equation has one repeated solution.
3. If the discriminant is less than 0 (D < 0), it means that the equation has no real solutions.
In this case, since the discriminant is 648 (D > 0), the equation has two distinct solutions, which means the correct answer is B) two solutions.