Factor the high-degree polynomial expression 7x2+42x+63=0
.(1 point)
x=
The expression can be factored as follows:
7(x^2 + 6x + 9) = 0
Now, we can factor the quadratic expression within the parentheses:
7(x + 3)(x + 3) = 0
So, the factorized form is:
(x + 3)^2 = 0
This means that the only solution is x = -3.
To factor the high-degree polynomial expression 7x^2 + 42x + 63 = 0, we can first divide the equation by 7 to simplify it:
x^2 + 6x + 9 = 0
Next, we can try to factor the quadratic expression x^2 + 6x + 9. Since the coefficient of x^2 is 1, we need to find two numbers whose product is 9 and sum is 6.
The numbers that satisfy this condition are 3 and 3, so we can factor the quadratic expression as:
(x + 3)(x + 3) = 0
Therefore, the factored form of the equation 7x^2 + 42x + 63 = 0 is:
(x + 3)(x + 3) = 0
To find the value of x, we set each factor equal to zero and solve:
x + 3 = 0
x = -3
So, the solution to the equation 7x^2 + 42x + 63 = 0 is x = -3.
To factor the polynomial expression 7x^2 + 42x + 63 = 0, we first check if it can be factored using the common factor. In this case, there is no common factor among the coefficients.
Next, we can try factoring by grouping. We look for two numbers that multiply to give the product of the coefficient of x^2 (7) and the constant term (63), which is 7 * 63 = 441. We also want these two numbers to add up to the coefficient of x (42).
The factors of 441 are 1 * 441, 3 * 147, 7 * 63, 9 * 49, 21 * 21. Among these, 7 * 63 = 441 satisfies our conditions, as 7 + 63 = 70.
So, we rewrite the middle term as 7x + 63x, and split it:
7x^2 + 7x + 63x + 63 = 0
Now, we take out the common factor from the first two terms and the common factor from the last two terms:
7x(x + 1) + 63(x + 1) = 0
Notice that both terms have a factor of (x + 1), so we can factor that out:
(x + 1)(7x + 63) = 0
Now we have factored the original polynomial expression as (x + 1)(7x + 63) = 0. To find the solutions, we set each factor equal to zero and solve for x:
x + 1 = 0 or 7x + 63 = 0
Solving for x in each equation, we get:
x = -1 or 7x = -63
Dividing both sides of the second equation by 7, we get:
x = -9
Therefore, the solutions to the equation 7x^2 + 42x + 63 = 0 are x = -1 and x = -9.