Factor the high-degree polynomial expression 7x2+42x+63=0
.
To factor the given expression, we can first factor out the greatest common factor, which is 7:
7(x^2 + 6x + 9) = 0
Next, we can factor the quadratic expression inside the parentheses:
7(x + 3)(x + 3) = 0
Since (x + 3) appears twice, we can write it as a squared term:
7(x + 3)^2 = 0
So, the factored form of the expression 7x^2 + 42x + 63 = 0 is 7(x + 3)^2 = 0.
To factor the high-degree polynomial expression 7x^2 + 42x + 63 = 0, follow these steps:
Step 1: Look for common factors (if any): In this case, there are no common factors other than 1.
Step 2: Since the coefficient of the x^2 term is not 1, we'll use the quadratic formula to find the roots. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the roots are given by x = (-b ± √(b^2 - 4ac)) / (2a).
In our case, a = 7, b = 42, and c = 63. Plugging these values into the quadratic formula, we get:
x = (-42 ± √(42^2 - 4(7)(63))) / (2(7))
Step 3: Simplify the expression inside the square root:
x = (-42 ± √(1764 - 1764)) / 14
x = (-42 ± √0) / 14
x = (-42 ± 0) / 14
Step 4: Simplify the expression:
x = -42/14 = -3
x = -42/14 = -3
Step 5: Write the factored form:
Since both values of x are the same (-3), the factored form of the polynomial is:
(x + 3)(x + 3) = 0
To factor the high-degree polynomial expression 7x^2 + 42x + 63 = 0, we will use the factoring method or the quadratic formula.
1. Factoring method:
We can first try to factor out common factors, but in this case, there aren't any common factors other than 1.
Next, we need to find two numbers whose product is 7 * 63 = 441 and whose sum is 42 (the coefficient of the x-term).
The factors of 441 are:
1 * 441
3 * 147
7 * 63
9 * 49
21 * 21
Since the sum we are looking for is 42, we see that 21 + 21 = 42.
Therefore, we can rewrite the expression as:
7x^2 + 21x + 21x + 63 = 0
Now, we can group the terms:
(7x^2 + 21x) + (21x + 63) = 0
And factor out the common factors:
7x(x + 3) + 21(x + 3) = 0
Notice that we have a common term, (x + 3), which we can factor out:
(x + 3)(7x + 21) = 0
Now, we have factored the polynomial expression as (x + 3)(7x + 21) = 0.
2. Quadratic formula:
If factoring doesn't work, you can always use the quadratic formula to find the roots of the polynomial expression.
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the given equation 7x^2 + 42x + 63 = 0, the coefficients are:
a = 7
b = 42
c = 63
Plugging these values into the quadratic formula, we get:
x = (-42 ± √(42^2 - 4 * 7 * 63)) / (2 * 7)
Simplifying further:
x = (-42 ± √(1764 - 1764)) / 14
x = (-42 ± √0) / 14
x = (-42 ± 0) / 14
We find that both roots are -3.
Therefore, the factored form of the polynomial expression 7x^2 + 42x + 63 = 0 is (x + 3)(7x + 21) = 0. The roots of the equation are x = -3.