Question

Factor the high-degree polynomial expression 7x2+42x+63=0.

x=__

Answer 1

To factor the polynomial expression \( 7x^2 + 42x + 63 = 0 \), we can start by factoring out the greatest common factor (GCF) from the polynomial. The GCF of the coefficients \( 7 \), \( 42 \), and \( 63 \) is \( 7 \).

1. Factor out \( 7 \):
\[
7(x^2 + 6x + 9) = 0
\]

2. Next, we need to factor the quadratic expression \( x^2 + 6x + 9 \). We can do this by looking for two numbers that multiply to \( 9 \) (the constant term) and add up to \( 6 \) (the coefficient of \( x \)). The numbers \( 3 \) and \( 3 \) work:
\[
x^2 + 6x + 9 = (x + 3)(x + 3) = (x + 3)^2
\]

3. Putting it all together, we have:
\[
7(x + 3)^2 = 0
\]

4. To find the roots of the equation, set the factored expression equal to zero:
\[
7(x + 3)^2 = 0
\]

5. Since \( 7 \neq 0 \), we can divide both sides by \( 7 \):
\[
(x + 3)^2 = 0
\]

6. Now, take the square root of both sides:
\[
x + 3 = 0
\]

7. Solving for \( x \):
\[
x = -3
\]

Thus, the solution to the equation \( 7x^2 + 42x + 63 = 0 \) is:
\[
x = -3
\]