Answer :
To solve the quadratic equation \( x(x + 6) = 0 \) numerically, we need to find the values of \( x \) which make the equation true (i.e., yield 0). Let’s break it down step-by-step:
1. Expand the Equation:
The equation is already factored as \( x(x + 6) = 0 \).
2. Apply the Zero Product Property:
According to the zero product property, if the product of two factors equals zero, then at least one of the factors must be zero. Thus, we have:
\( x = 0 \)
or
\( x + 6 = 0 \).
3. Solve Each Factor for Zero:
Solve the equation \( x = 0 \):
[tex]\[ x = 0 \][/tex]
Solve the equation \( x + 6 = 0 \):
[tex]\[ x + 6 = 0 \implies x = -6 \][/tex]
4. Identify the Solutions:
The solutions to the quadratic equation \( x(x + 6) = 0 \) are:
[tex]\[ x = 0 \quad \text{or} \quad x = -6 \][/tex]
5. Examine the Choices Provided:
- \( a. \ x = -1 \ \text{or} \ x = 3 \)
- \( b. \ x = 0 \ \text{or} \ x = 6 \)
- \( c. \ x = 0 \ \text{or} \ x = -3 \)
- \( d. \ x = 0 \ \text{or} \ x = -6 \)
6. Select the Correct Answer:
The correct choice is:
[tex]\[ \boxed{d. \ x = 0 \ \text{or} \ x = -6} \][/tex]
Thus, the quadratic equation [tex]\( x(x + 6) = 0 \)[/tex] is satisfied when [tex]\( x = 0 \)[/tex] or [tex]\( x = -6 \)[/tex], making choice D the correct answer.
1. Expand the Equation:
The equation is already factored as \( x(x + 6) = 0 \).
2. Apply the Zero Product Property:
According to the zero product property, if the product of two factors equals zero, then at least one of the factors must be zero. Thus, we have:
\( x = 0 \)
or
\( x + 6 = 0 \).
3. Solve Each Factor for Zero:
Solve the equation \( x = 0 \):
[tex]\[ x = 0 \][/tex]
Solve the equation \( x + 6 = 0 \):
[tex]\[ x + 6 = 0 \implies x = -6 \][/tex]
4. Identify the Solutions:
The solutions to the quadratic equation \( x(x + 6) = 0 \) are:
[tex]\[ x = 0 \quad \text{or} \quad x = -6 \][/tex]
5. Examine the Choices Provided:
- \( a. \ x = -1 \ \text{or} \ x = 3 \)
- \( b. \ x = 0 \ \text{or} \ x = 6 \)
- \( c. \ x = 0 \ \text{or} \ x = -3 \)
- \( d. \ x = 0 \ \text{or} \ x = -6 \)
6. Select the Correct Answer:
The correct choice is:
[tex]\[ \boxed{d. \ x = 0 \ \text{or} \ x = -6} \][/tex]
Thus, the quadratic equation [tex]\( x(x + 6) = 0 \)[/tex] is satisfied when [tex]\( x = 0 \)[/tex] or [tex]\( x = -6 \)[/tex], making choice D the correct answer.
