Question

12x² + 27x = -6
p: s:
signs:
factor:
solve:

Explanation:

Step1: Rewrite the equation

First, we rewrite the equation \(12x^{2}+27x=-6\) in standard quadratic form \(ax^{2}+bx + c = 0\) by adding 6 to both sides:
\(12x^{2}+27x + 6=0\)
We can simplify this equation by dividing all terms by 3:
\(4x^{2}+9x + 2 = 0\)

Step2: Identify \(a\), \(b\), \(c\) for factoring

For the quadratic equation \(ax^{2}+bx + c=0\) (here \(a = 4\), \(b=9\), \(c = 2\)), we need to find two numbers that multiply to \(a\times c=4\times2 = 8\) (this is \(P\)) and add up to \(b = 9\) (this is \(S\)). The two numbers are 8 and 1 since \(8\times1=8\) and \(8 + 1=9\). The signs of both numbers are positive (since \(a\times c\) and \(b\) are positive).

Step3: Factor the quadratic

We rewrite the middle term using the two numbers we found:
\(4x^{2}+8x+x + 2=0\)
Group the terms:
\((4x^{2}+8x)+(x + 2)=0\)
Factor out the greatest common factor from each group:
\(4x(x + 2)+1(x + 2)=0\)
Now factor out \((x + 2)\):
\((4x + 1)(x + 2)=0\)

Step4: Solve for \(x\)

Set each factor equal to zero and solve:

• For \(4x+1=0\):

Subtract 1 from both sides: \(4x=-1\)
Divide by 4: \(x=-\frac{1}{4}\)

• For \(x + 2=0\):

Subtract 2 from both sides: \(x=-2\)

Answer:

• \(P\): \(8\), \(S\): \(9\), Signs: Both positive.
• Factor: \((4x + 1)(x + 2)\)
• Solve: \(x=-\frac{1}{4}\) or \(x=-2\)