To solve the equation [tex]\( x + \frac{35}{x} = -12 \)[/tex], let's find the values of [tex]\( x \)[/tex] that satisfy this equation.
1. Rearrange the Equation:
Start by moving all terms to one side of the equation to set it to zero:
[tex]\[
x + \frac{35}{x} + 12 = 0
\][/tex]
2. Multiply by [tex]\( x \)[/tex]:
To clear the fraction, multiply every term by [tex]\( x \)[/tex]:
[tex]\[
x^2 + 35 + 12x = 0
\][/tex]
3. Form a Quadratic Equation:
Combine like terms:
[tex]\[
x^2 + 12x + 35 = 0
\][/tex]
4. Solve the Quadratic Equation:
Recognize that this is a standard quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex]. To solve it, either factorize it or use the quadratic formula.
5. Apply the Quadratic Formula:
The quadratic formula is:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
For our equation [tex]\( x^2 + 12x + 35 = 0 \)[/tex], [tex]\(a = 1\)[/tex], [tex]\(b = 12\)[/tex], and [tex]\(c = 35\)[/tex].
6. Calculate the Discriminant:
[tex]\[
\Delta = b^2 - 4ac = 12^2 - 4 \cdot 1 \cdot 35 = 144 - 140 = 4
\][/tex]
7. Find the Solutions:
Now, calculate the values of [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-12 \pm \sqrt{4}}{2 \cdot 1} = \frac{-12 \pm 2}{2}
\][/tex]
This gives us two solutions:
[tex]\[
x = \frac{-12 + 2}{2} = \frac{-10}{2} = -5
\][/tex]
[tex]\[
x = \frac{-12 - 2}{2} = \frac{-14}{2} = -7
\][/tex]
Therefore, the solutions to the equation [tex]\( x + \frac{35}{x} = -12 \)[/tex] are:
[tex]\[
x = [-7, -5]
\][/tex]
