Simplify Rational Equation
Begin with the given rational equation:
$ \frac{x-3}{x-4} + \frac{x-6}{x-5} = \frac{5}{2} $
Combine fractions on the left side using a common denominator:
$ \frac{(x-3)(x-5) + (x-6)(x-4)}{(x-4)(x-5)} = \frac{5}{2} $
Expand and simplify the numerator and denominator:
- Numerator: $ (x^2 - 8x + 15) + (x^2 - 10x + 24) = 2x^2 - 18x + 39 $
- Denominator: $ x^2 - 9x + 20 $
Substitute back into the equation:
$ \frac{2x^2 - 18x + 39}{x^2 - 9x + 20} = \frac{5}{2} $
Cross-multiply:
$ 2(2x^2 - 18x + 39) = 5(x^2 - 9x + 20) $
$ 4x^2 - 36x + 78 = 5x^2 - 45x + 100 $
Rearrange into standard quadratic form $ ax^2 + bx + c = 0 $:
$ 0 = x^2 - 9x + 22 $
Calculate Discriminant
Identify the coefficients $a$, $b$, and $c$ from the quadratic equation $ x^2 - 9x + 22 = 0 $:
- $ a = 1 $
- $ b = -9 $
- $ c = 22 $
Apply the discriminant formula $ \Delta = b^2 - 4ac $:
$ \Delta = (-9)^2 - 4(1)(22) $
$ \Delta = 81 - 88 $
$ \Delta = -7 $
Important Questions from Quadratic Equation
If 2x 2+ 5x + 1 = 0, then one of the values of \(x - \frac{1}{{2x}}\) is:
If \(a-\frac{12}{a}=1\) , where a > 0, then the value of \(a^2+\frac{16}{a^2}\) is:
If x 2 – 3x + 1 = 0, then the value of \(\frac{(x^4+\frac{1}{x^2})}{(x^2+5x+1)}\) is:
If \(\sqrt{x}{}-{1\over\sqrt{x}}=\sqrt5\) , \(x \ne 0\) , then what is the value of \((x^4+{1\over{x^2}})\over(x^2+1) \) ?
If x 2 + \(\frac{1}{x^2}\) = 18, x > 0, then find the value of x 3 + \(\frac{1}{x^3}\) .
