Q. \(\;3x^2 + 30x + 75 =\;\)

Q: What is the quadratic equation 3x^2+30x+75=0 factored as?

3x^2+30x+75=3(x^2+10x+25)=3(x+5)^2.

Q: What is the discriminant of 3x^2+30x+75=0?

a=3,b=30,c=75. \Delta=b^2-4ac=900-900=0.

Q: Solve 3x^2+30x+75=0 for x.

Since (x+5)^2=0, x=-5 (double root).

Q: How do you solve 3x^2+30x+75=0 using the quadratic formula?

x=\frac{-30\pm\sqrt{0}}{2\cdot3}=\frac{-30}{6}=-5.

Q: What is the vertex of y=3x^2+30x+75?

x_v=-\frac{b}{2a}=-\frac{30}{6}=-5. y_v=f(-5)=0.

Q: Can you complete the square for 3x^2+30x+75?

3(x^2+10x+25)=3(x+5)^2, because x^2+10x+25=(x+5)^2.

Answer

Factor the quadratic:

\[
3x^2+30x+75=3(x^2+10x+25)=3(x+5)^2
\]
So the expression equals \(3(x+5)^2\).

Detailed Explanation

We want to solve the equation:

\[
3x^2 + 30x + 75 = 0
\]

Step 1: Check whether the quadratic can be factored.

Notice every term is divisible by 3:

\[
3x^2 + 30x + 75 = 3(x^2 + 10x + 25)
\]

So the equation becomes:

\[
3(x^2 + 10x + 25)=0
\]

Step 2: Find roots by setting the factor equal to zero.

Since \(3 \neq 0\), the product is zero only if the other factor is zero:

\[
x^2 + 10x + 25 = 0
\]

Step 3: Factor the trinomial as a perfect square.

We check whether \(x^2 + 10x + 25\) matches the form \((x+a)^2\).

Recall:

\[
(x+a)^2 = x^2 + 2ax + a^2
\]

Match coefficients:

\(2a = 10\) so \(a = 5\)
\(a^2 = 25\), which matches the constant term

Therefore, the trinomial is a perfect square:

\[
x^2 + 10x + 25 = (x+5)^2
\]

Step 4: Set the squared factor equal to zero.

\[
(x+5)^2 = 0
\]

Step 5: Take the square root of both sides.

The square is zero only when the inside is zero:

\[
x+5 = 0
\]

Step 6: Solve for \(x\).

\[
x = -5
\]

Final Answer:

\[
x = -5
\]

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Algebra FAQ

What is the quadratic equation \(3x^2+30x+75=0\) factored as?

\(3x^2+30x+75=3(x^2+10x+25)=3(x+5)^2\).

What is the discriminant of \(3x^2+30x+75=0\)?

\(a=3,b=30,c=75\). \(\Delta=b^2-4ac=900-900=0\).

Solve \(3x^2+30x+75=0\) for \(x\).

Since \((x+5)^2=0\), \(x=-5\) (double root).

How do you solve \(3x^2+30x+75=0\) using the quadratic formula?

\(x=\frac{-30\pm\sqrt{0}}{2\cdot3}=\frac{-30}{6}=-5\).

What is the vertex of \(y=3x^2+30x+75\)?

\(x_v=-\frac{b}{2a}=-\frac{30}{6}=-5\). \(y_v=f(-5)=0\).

Can you complete the square for \(3x^2+30x+75\)?

\(3(x^2+10x+25)=3(x+5)^2\), because \(x^2+10x+25=(x+5)^2\).

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