To solve the equation [tex]\( \frac{1}{3(x+7)} - \frac{1}{2(x+1)} = -6 \)[/tex], where [tex]\( x \)[/tex] represents the number of months, follow these steps:
1. Rewrite the equation:
[tex]\[
\frac{1}{3(x+7)} - \frac{1}{2(x+1)} = -6
\][/tex]
2. Find a common denominator for the fractions on the left so that you can combine them. The terms are [tex]\( 3(x+7) \)[/tex] and [tex]\( 2(x+1) \)[/tex], so the common denominator is [tex]\( 6(x+7)(x+1) \)[/tex].
3. Rewrite each fraction with the common denominator:
[tex]\[
\frac{2(x+1)}{6(x+7)(x+1)} - \frac{3(x+7)}{6(x+7)(x+1)} = -6
\][/tex]
4. Combine the fractions:
[tex]\[
\frac{2(x+1) - 3(x+7)}{6(x+7)(x+1)} = -6
\][/tex]
5. Simplify the numerator:
[tex]\[
2(x + 1) - 3(x + 7) = 2x + 2 - 3x - 21 = -x - 19
\][/tex]
So, the equation becomes:
[tex]\[
\frac{-x - 19}{6(x+7)(x+1)} = -6
\][/tex]
6. Eliminate the denominator by multiplying both sides of the equation by [tex]\( 6(x+7)(x+1) \)[/tex]:
[tex]\[
-x - 19 = -6 \cdot 6(x+7)(x+1)
\][/tex]
[tex]\[
-x - 19 = -36(x^2 + 8x + 7)
\][/tex]
7. Simplify the right side:
[tex]\[
-x - 19 = -36x^2 - 288x - 252
\][/tex]
8. Move all terms to one side to form a standard quadratic equation:
[tex]\[
-36x^2 - 288x - 252 + x + 19 = 0
\][/tex]
[tex]\[
-36x^2 - 287x - 233 = 0
\][/tex]
Multiply through by [tex]\(-1\)[/tex] to make the coefficients positive:
[tex]\[
36x^2 + 287x + 233 = 0
\][/tex]
9. Solve the quadratic equation: [tex]\( 36x^2 + 287x + 233 = 0 \)[/tex]. To do this, we use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 36 \)[/tex], [tex]\( b = 287 \)[/tex], and [tex]\( c = 233 \)[/tex].
The solutions given in the result are already simplified and are:
[tex]\[
x = \frac{-287 - \sqrt{48817}}{72} \quad \text{and} \quad x = \frac{-287 + \sqrt{48817}}{72}
\][/tex]
Therefore, the number of months Michael will have to pay on the car corresponds to the roots of this equation, which are:
[tex]\[
x = \frac{-287 - \sqrt{48817}}{72} \quad \text{and} \quad x = \frac{-287 + \sqrt{48817}}{72}
\][/tex]
