Solving Quadratic Equations with Fractional Solutions: Methods and Techniques
Quadratic equations with fractional solutions can be approached using the same methods as those used for integer, real number, or decimal solutions. Whether you are solving for x with a fractional value or an integer, the fundamental techniques of factoring, completing the square, and using the quadratic formula remain consistent.
Overview of Solving Methods
Similar to solving quadratic equations with other values of x, the primary methods for solving quadratic equations with fractional solutions are factoring, completing the square, and using the quadratic formula. The solutions obtained are the values of x obtained after applying these methods correctly.
Example: Factoring Quadratic Equations with Fractional Solutions
Consider the equation 15x^2 x - 6 0. This can be factored as (3x 2)(5x - 3) 0.
To solve for x, set each factor equal to zero: 3x 2 0 gives x -frac{2}{3} 5x - 3 0 gives x frac{3}{5}Removing Fractions in Quadratic Equations
Sometimes, the coefficients of a quadratic equation contain fractions, which can complicate the solving process. One effective method to simplify these equations is to multiply each term by the least common multiple (LCM) of the denominators of the coefficients.
Example 1: Solving with Fractions
Consider the equation frac{2}{3}x^2 - frac{5}{6}x - 1 0. The LCM of the denominators (3 and 6) is 6. Multiply every term by 6:
6 cdot frac{2}{3}x^2 - 6 cdot frac{5}{6}x - 6 cdot 1 0 Rightarrow 4x^2 - 5x - 6 0
To solve this, use the quadratic formula:
x frac{-(-5) pm sqrt{(-5)^2 - 4 cdot 4 cdot (-6)}}{2 cdot 4}
x frac{5 pm sqrt{25 96}}{8} frac{5 pm sqrt{121}}{8} frac{5 pm 11}{8}
This gives two solutions: x frac{16}{8} 2 and x frac{-6}{8} -frac{3}{4}.
Completing the Square
Another method to solve quadratic equations is completing the square. This method is especially useful when the quadratic equation has fractional coefficients, as demonstrated in the following example.
Example 2: Using Completing the Square with Fractions
Consider the equation frac{1}{2}x^2 - frac{3}{4}x - frac{2}{3} 0. The LCM of the denominators (2, 4, and 3) is 12.
Multiply each term by 12:
12 cdot frac{1}{2}x^2 - 12 cdot frac{3}{4}x - 12 cdot frac{2}{3} 0 Rightarrow 6x^2 - 9x - 8 0
Now, complete the square on the terms involving x:
6(x^2 - frac{3}{2}x) - 8 0 Rightarrow 6(x^2 - frac{3}{2}x frac{9}{16} - frac{9}{16}) - 8 0 Rightarrow 6((x - frac{3}{4})^2 - frac{9}{16}) - 8 0
6(x - frac{3}{4})^2 - frac{54}{16} - 8 0 Rightarrow 6(x - frac{3}{4})^2 - frac{54}{16} - frac{128}{16} 0 Rightarrow 6(x - frac{3}{4})^2 frac{182}{16} Rightarrow (x - frac{3}{4})^2 frac{182}{96} Rightarrow (x - frac{3}{4})^2 frac{91}{48}
x - frac{3}{4} pm sqrt{frac{91}{48}} Rightarrow x frac{3}{4} pm sqrt{frac{91}{48}}
Therefore, the solutions are x frac{3}{4} sqrt{frac{91}{48}} and x frac{3}{4} - sqrt{frac{91}{48}}.
Using the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations, even when the solutions are fractions. The formula is given by:
x frac{-b pm sqrt{b^2 - 4ac}}{2a}
where a, b, and c are the coefficients of the quadratic equation ax^2 bx c 0.
Example 3: Applying the Quadratic Formula
Consider the equation frac{1}{3}x^2 - frac{2}{5}x frac{1}{6} 0. The coefficients are a frac{1}{3}, b -frac{2}{5}, and c frac{1}{6}.
Substitute the values into the quadratic formula:
x frac{-left(-frac{2}{5}right) pm sqrt{left(-frac{2}{5}right)^2 - 4 cdot frac{1}{3} cdot frac{1}{6}}}{2 cdot frac{1}{3}}
x frac{frac{2}{5} pm sqrt{frac{4}{25} - frac{4}{18}}}{frac{2}{3}}
x frac{frac{2}{5} pm sqrt{frac{72 - 100}{225}}}{frac{2}{3}}
x frac{frac{2}{5} pm sqrt{frac{-28}{225}}}{frac{2}{3}}
x frac{frac{2}{5} pm sqrt{frac{-28}{225}} cdot frac{3}{2}}{frac{2}{3}}
x frac{frac{2}{5} pm frac{3isqrt{28}}{30}}{frac{2}{3}}
x frac{frac{2}{5} pm frac{isqrt{28}}{10}}{frac{2}{3}}
Therefore, the solutions are complex: x frac{3}{5} pm frac{3isqrt{28}}{10}.
Conclusion
In conclusion, solving quadratic equations with fractional solutions involves the same fundamental methods as solving equations with other types of values for x. Whether you choose to factor, complete the square, or use the quadratic formula, each method requires careful manipulation of the coefficients and proper application of mathematical principles. By mastering these techniques, you can confidently solve a wide range of quadratic equations, even those with fractional coefficients.