Answer:
Check the step by step process below.
Step-by-step explanation:
A Rational Equation is an equation composed of rational expressions. A rational expression is in the form of [tex]\frac{f(x)}{q(x)}[/tex] where [tex]q(x)[/tex] is not equal to zero.
A Quadratic Equation is in the form of [tex]ax^2+bx+c=0[/tex] where [tex]a[/tex], [tex]b[/tex], and [tex]c[/tex] are constants but [tex]a\ne0[/tex].
Rational Equation that can be transformed into Quadratic Equation
1. It has to be a rational equation.
2. There should only be one variable.
3. If the expressions are linear, then there should be a variable on both numerator and denominator.
4. If the denominators are all constant, then the numerator should have at least one variable that has a power of 2.
Steps in Transforming a Rational Equation Algebraic Equation into a Quadratic Equation
1. Multiply both sides of the equation by the LCD.
2. Apply the distributive law.
3. Combine like terms.
4. Apply the inverse operation.
5. Rewrite the equation in the form [tex]ax^2+bx+c=0[/tex].
Examples:
In relation to item 3 above. (If the expressions are linear, then there should be a variable on both numerator and denominator. )
[tex]\begin{aligned}\frac{1+x}{x}+\frac{3x}{x+1}&=-4\\x(x+1)\left(\frac{1+x}{x}+\frac{3x}{x+1}\right)&=x(x+1)(-4)\\\frac{x(x+1)(1+x)}{x}+\frac{x(x+1)(3x)}{x+1}&=-4x^2-4x\\(x+1)(1+x)+x(3x)&=-4x^2-4x\\x^2+2x+1+3x^2&=-4x^2-4x\\4x^2+2x+1&=-4x^2-4x\\4x^2+4x^2+2x+4x+1&=-4x^2+4x^2-4x+4x\\8x^2+6x+1&=0\end{aligned}[/tex]
In relation to item 4 above. ( If the denominators are all constant, then the numerator should have at least one variable that has a power of 2. )
[tex]\begin{aligned}\frac{x^2+1}{3}+\frac{4x+6}{5}&=\frac{6}{7}\\105\left(\frac{x^2+1}{3}+\frac{4x+6}{5}\right)&=105\left(\frac{6}{7}\right)\\\frac{105(x^2+1)}{3}+\frac{105(4x+6)}{5}&=90\\35(x^2+1)+21(4x+6)&=90\\35x^2+35+84x+126&=90\\35x^2+84x+161&=90\\35x^2+84x+161-90&=90-90\\35x^2+84x+71&=0\end{aligned}[/tex]
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