To solve the given equation [tex]\(\frac{x^2 - x - 6}{x^2} = \frac{x - 6}{2x} + \frac{2x + 12}{x}\)[/tex], we start by finding a common denominator and simplifying. After these operations, the equation simplifies to:
[tex]\[ 3x^2 - 20x + 12 = 0 \][/tex]
This is a quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex] where [tex]\(a = 3\)[/tex], [tex]\(b = -20\)[/tex], and [tex]\(c = 12\)[/tex].
To solve this quadratic equation, we can use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
1. First, we calculate the discriminant ([tex]\(\Delta\)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values:
[tex]\[ \Delta = (-20)^2 - 4 \cdot 3 \cdot 12 \][/tex]
[tex]\[ \Delta = 400 - 144 \][/tex]
[tex]\[ \Delta = 256 \][/tex]
2. Since the discriminant is positive ([tex]\(\Delta = 256\)[/tex]), the quadratic equation has two real and distinct solutions, which we will now find:
[tex]\[ x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
3. Substitute the known values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex]:
[tex]\[ x_{1,2} = \frac{-(-20) \pm \sqrt{256}}{2 \cdot 3} \][/tex]
[tex]\[ x_{1,2} = \frac{20 \pm 16}{6} \][/tex]
Now, we calculate the two solutions separately:
4. For the positive square root:
[tex]\[ x_1 = \frac{20 + 16}{6} \][/tex]
[tex]\[ x_1 = \frac{36}{6} \][/tex]
[tex]\[ x_1 = 6 \][/tex]
5. For the negative square root:
[tex]\[ x_2 = \frac{20 - 16}{6} \][/tex]
[tex]\[ x_2 = \frac{4}{6} \][/tex]
[tex]\[ x_2 = \frac{2}{3} \][/tex]
[tex]\[ x_2 \approx 0.6666666666666666 \][/tex]
Thus, the solutions to the equation [tex]\(3x^2 - 20x + 12 = 0\)[/tex] are [tex]\( x_1 = 6 \)[/tex] and [tex]\( x_2 \approx 0.6666666666666666 \)[/tex].
In conclusion:
The solutions to the equation are:
[tex]\[ x_1 = 6 \][/tex]
[tex]\[ x_2 = \frac{2}{3} \][/tex] and the discriminant is [tex]\(\Delta = 256\)[/tex].
