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Answer :
Sure! Let's solve the quadratic equation step by step by factoring:
1. Start with the equation:
[tex]\( 3x^2 = 27 \)[/tex].
2. Rewrite it in standard quadratic form:
To do this, subtract 27 from both sides:
[tex]\[ 3x^2 - 27 = 0 \][/tex].
3. Factor out the common factor:
Notice that both terms on the left side have a common factor of 3. Factor 3 out:
[tex]\[ 3(x^2 - 9) = 0 \][/tex].
4. Further factor the quadratic expression:
The expression inside the parenthesis, [tex]\( x^2 - 9 \)[/tex], is a difference of squares. It can be factored into:
[tex]\[ (x + 3)(x - 3) \][/tex].
5. Rewrite the equation:
The equation becomes:
[tex]\[ 3(x + 3)(x - 3) = 0 \][/tex].
6. Apply the Zero Product Property:
According to the Zero Product Property, if the product of several factors equals zero, at least one of the factors must be zero. Thus, set each factor to zero:
[tex]\[ x + 3 = 0 \][/tex] or [tex]\[ x - 3 = 0 \][/tex].
7. Solve each equation for [tex]\( x \)[/tex]:
- For [tex]\( x + 3 = 0 \)[/tex], subtract 3 from both sides:
[tex]\[ x = -3 \][/tex].
- For [tex]\( x - 3 = 0 \)[/tex], add 3 to both sides:
[tex]\[ x = 3 \][/tex].
So, the solutions to the equation are [tex]\( x = -3 \)[/tex] and [tex]\( x = 3 \)[/tex].
1. Start with the equation:
[tex]\( 3x^2 = 27 \)[/tex].
2. Rewrite it in standard quadratic form:
To do this, subtract 27 from both sides:
[tex]\[ 3x^2 - 27 = 0 \][/tex].
3. Factor out the common factor:
Notice that both terms on the left side have a common factor of 3. Factor 3 out:
[tex]\[ 3(x^2 - 9) = 0 \][/tex].
4. Further factor the quadratic expression:
The expression inside the parenthesis, [tex]\( x^2 - 9 \)[/tex], is a difference of squares. It can be factored into:
[tex]\[ (x + 3)(x - 3) \][/tex].
5. Rewrite the equation:
The equation becomes:
[tex]\[ 3(x + 3)(x - 3) = 0 \][/tex].
6. Apply the Zero Product Property:
According to the Zero Product Property, if the product of several factors equals zero, at least one of the factors must be zero. Thus, set each factor to zero:
[tex]\[ x + 3 = 0 \][/tex] or [tex]\[ x - 3 = 0 \][/tex].
7. Solve each equation for [tex]\( x \)[/tex]:
- For [tex]\( x + 3 = 0 \)[/tex], subtract 3 from both sides:
[tex]\[ x = -3 \][/tex].
- For [tex]\( x - 3 = 0 \)[/tex], add 3 to both sides:
[tex]\[ x = 3 \][/tex].
So, the solutions to the equation are [tex]\( x = -3 \)[/tex] and [tex]\( x = 3 \)[/tex].
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