Use factoring to solve the polynomial equation:
HINT: Try multiplying each side of the equation by a number to eliminate the denominators.
x2/162 + X/18 + 1/9 =0
I will assume you meant:
x^2/162 +x/18 + 1/9 = 0
the LCD is 162, so multiply each term by that
x^2 + 9x + 18 = 0
(x+3)(x+6) = 0
x = -3 or x = -6
To solve the polynomial equation x^2/162 + x/18 + 1/9 = 0 using factoring, the first step is to multiply each side of the equation by the least common multiple of the denominators to eliminate the fractions.
The denominators in this equation are 162, 18, and 9. To find the least common multiple (LCM) of these numbers, we can start by finding the prime factorization of each number:
162 = 2 × 3 × 3 × 3 × 3 = 2 × 3^4
18 = 2 × 3 × 3 = 2 × 3^2
9 = 3 × 3 = 3^2
To find the LCM, we take the highest power of each prime factor that appears in any of the numbers:
LCM = 2 × 3^4 = 162
Now, let's multiply each term of the equation by 162:
162 * (x^2/162) + 162 * (x/18) + 162 * (1/9) = 0
Simplifying, we get:
x^2 + 9x + 18 = 0
Now, the equation is in a form that can be factored. We want to find two binomial expressions that, when multiplied, equal the quadratic equation:
(x + a)(x + b) = 0
By expanding this expression, we get:
x^2 + (a+b)x + ab = 0
We want the coefficients of x^2, x, and the constant term to match our original equation x^2 + 9x + 18 = 0.
From the constant term (18), we can see that the factors should be of the form (x + c)(x + d), where c and d are integers that multiply to give 18. Possible factors of 18 are (1, 18), (2, 9), and (3, 6).
From the coefficient of x (9), we want c + d to be equal to 9. Among the possible factors of 18, the only pair that satisfies this condition is (2, 9).
So we rewrite the quadratic equation using these factors:
(x + 2)(x + 9) = 0
Now, we can solve for x by setting each factor equal to zero:
x + 2 = 0 --> x = -2
x + 9 = 0 --> x = -9
So the solutions to the equation are x = -2 and x = -9.