Question
Factor completely.
\[3 x^2 +30 x +75=\]
\[3 x^2 +30 x +75=\]
Answers
Answered by
GPT-4o mini
To factor the expression \(3 x^2 + 30 x + 75\), we start by noticing that we can factor out the greatest common factor (GCF) from all the terms.
1. The GCF of the coefficients \(3\), \(30\), and \(75\) is \(3\). So, we factor out \(3\):
\[
3(x^2 + 10x + 25)
\]
2. Next, we need to factor the quadratic expression \(x^2 + 10x + 25\). To do this, we look for two numbers that multiply to \(25\) (the constant term) and add to \(10\) (the coefficient of \(x\)). The numbers \(5\) and \(5\) work because:
\[
5 \cdot 5 = 25 \quad \text{and} \quad 5 + 5 = 10
\]
3. This means that we can rewrite the quadratic as a perfect square:
\[
x^2 + 10x + 25 = (x + 5)(x + 5) = (x + 5)^2
\]
4. Therefore, we can rewrite the entire expression:
\[
3(x^2 + 10x + 25) = 3((x + 5)^2)
\]
5. Finally, we put everything together:
\[
3(x + 5)^2
\]
Thus, the completely factored form of the expression \(3 x^2 + 30 x + 75\) is:
\[
\boxed{3(x + 5)^2}
\]
1. The GCF of the coefficients \(3\), \(30\), and \(75\) is \(3\). So, we factor out \(3\):
\[
3(x^2 + 10x + 25)
\]
2. Next, we need to factor the quadratic expression \(x^2 + 10x + 25\). To do this, we look for two numbers that multiply to \(25\) (the constant term) and add to \(10\) (the coefficient of \(x\)). The numbers \(5\) and \(5\) work because:
\[
5 \cdot 5 = 25 \quad \text{and} \quad 5 + 5 = 10
\]
3. This means that we can rewrite the quadratic as a perfect square:
\[
x^2 + 10x + 25 = (x + 5)(x + 5) = (x + 5)^2
\]
4. Therefore, we can rewrite the entire expression:
\[
3(x^2 + 10x + 25) = 3((x + 5)^2)
\]
5. Finally, we put everything together:
\[
3(x + 5)^2
\]
Thus, the completely factored form of the expression \(3 x^2 + 30 x + 75\) is:
\[
\boxed{3(x + 5)^2}
\]
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