Answer :
To determine all the real roots of the equation [tex]\((x+7)\left(x^2-49\right)=0\)[/tex], we proceed as follows:
1. Recognize that the equation is a product of two factors: [tex]\((x+7)\)[/tex] and [tex]\((x^2-49)\)[/tex].
2. To find the roots of the equation, we set each factor equal to zero since a product is zero if and only if at least one of the factors is zero.
3. Solve the first factor:
[tex]\[
x + 7 = 0
\][/tex]
Subtract 7 from both sides:
[tex]\[
x = -7
\][/tex]
4. Solve the second factor:
[tex]\[
x^2 - 49 = 0
\][/tex]
Add 49 to both sides:
[tex]\[
x^2 = 49
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
x = \pm 7
\][/tex]
So, [tex]\(x\)[/tex] can be either 7 or -7.
Combining the solutions from both factors, the real roots of the equation [tex]\((x+7)(x^2-49)=0\)[/tex] are:
[tex]\[
x = -7, 7
\][/tex]
Therefore, the correct choices are:
C. -7
1. 7
1. Recognize that the equation is a product of two factors: [tex]\((x+7)\)[/tex] and [tex]\((x^2-49)\)[/tex].
2. To find the roots of the equation, we set each factor equal to zero since a product is zero if and only if at least one of the factors is zero.
3. Solve the first factor:
[tex]\[
x + 7 = 0
\][/tex]
Subtract 7 from both sides:
[tex]\[
x = -7
\][/tex]
4. Solve the second factor:
[tex]\[
x^2 - 49 = 0
\][/tex]
Add 49 to both sides:
[tex]\[
x^2 = 49
\][/tex]
Taking the square root of both sides, we get:
[tex]\[
x = \pm 7
\][/tex]
So, [tex]\(x\)[/tex] can be either 7 or -7.
Combining the solutions from both factors, the real roots of the equation [tex]\((x+7)(x^2-49)=0\)[/tex] are:
[tex]\[
x = -7, 7
\][/tex]
Therefore, the correct choices are:
C. -7
1. 7
