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Whenever you solve a rational equation you must make sure the result obtained for an answer does not allow the denominator of one of the rational expressions to assume a value of ZERO, as division by zero is undefined and therefore prohibited. For example if we have 2x/(x-3) =(x2 -9x)/ x when we multiply out by x(x-3) we get 2x(x) = (x2 -9x)(x-3) so 2x2 = x(x-9)(x-3) 2x2 = x(x2 - 12x + 27) 2x2 = x3 - 12x2 + 27x so 0 = x3 - 14x2 + 27x 0 = x(x2 - 14x + 27) so solutions are 0 and 7 + √22 and 7 -√22 but 0 makes right hand side expression have zero in denominator so it is not a solution. We actually have to look at all obtained solutions to be sure they ae not extraneous. Suppose we had obtained a 3 for a solution. That would make the left side denominator equal zero and we would have to dismiss that, if 3 was obtained. The two irrational solutions we have obtained are genuine solutions as neither introduces a zero to a denominator
What else can I help you with?
How can you determine whether a polynomial equation has imaginary solutions?
To determine whether a polynomial equation has imaginary solutions, you must first identify what type of equation it is. If it is a quadratic equation, you can use the quadratic formula to solve for the solutions. If the equation is a cubic or higher order polynomial, you can use the Rational Root Theorem to determine if there are any imaginary solutions. The Rational Root Theorem states that if a polynomial equation has rational solutions, they must be a factor of the constant term divided by a factor of the leading coefficient. If there are no rational solutions, then the equation has imaginary solutions. To use the Rational Root Theorem, first list out all the possible rational solutions. Then, plug each possible rational solution into the equation and see if it is a solution. If there are any solutions, then the equation has imaginary solutions. If not, then there are no imaginary solutions.
What is an equation that has two radical expressions and no real roots?
It is simply an equation with non-rational solutions. There is no special name for it.
How is an extraneous solution of a ration equation similar to an excluded value of a rational equation?
An extraneous solution of a rational equation is a solution that emerges from the algebraic process but does not satisfy the original equation, while an excluded value is a value that makes the denominator zero and is therefore not permissible in the equation. Both concepts highlight the limitations and constraints of rational expressions. Excluded values can lead to extraneous solutions if they are mistakenly included in the solution set. Thus, both are essential to consider when solving rational equations to ensure valid solutions.
What is true of the discriminant when the two real number solutions to a quadratic equation are rational numbers?
The discriminant must be a perfect square or a square of a rational number.
Do all rational equations have a single solution?
Not all rational equations have a single solution but can have more than one because of having polynomials. All rational equations do have solutions that cannot fulfill the answer.
How can you determine whether a polynomial equation has imaginary solutions?
To determine whether a polynomial equation has imaginary solutions, you must first identify what type of equation it is. If it is a quadratic equation, you can use the quadratic formula to solve for the solutions. If the equation is a cubic or higher order polynomial, you can use the Rational Root Theorem to determine if there are any imaginary solutions. The Rational Root Theorem states that if a polynomial equation has rational solutions, they must be a factor of the constant term divided by a factor of the leading coefficient. If there are no rational solutions, then the equation has imaginary solutions. To use the Rational Root Theorem, first list out all the possible rational solutions. Then, plug each possible rational solution into the equation and see if it is a solution. If there are any solutions, then the equation has imaginary solutions. If not, then there are no imaginary solutions.
If the discriminant of an equation is negative?
It has two complex solutions.
What is an equation that has two radical expressions and no real roots?
It is simply an equation with non-rational solutions. There is no special name for it.
How is an extraneous solution of a ration equation similar to an excluded value of a rational equation?
An extraneous solution of a rational equation is a solution that emerges from the algebraic process but does not satisfy the original equation, while an excluded value is a value that makes the denominator zero and is therefore not permissible in the equation. Both concepts highlight the limitations and constraints of rational expressions. Excluded values can lead to extraneous solutions if they are mistakenly included in the solution set. Thus, both are essential to consider when solving rational equations to ensure valid solutions.
What is true of the discriminant when the two real number solutions to a quadratic equation are rational numbers?
The discriminant must be a perfect square or a square of a rational number.
Do all rational equations have a single solution?
Not all rational equations have a single solution but can have more than one because of having polynomials. All rational equations do have solutions that cannot fulfill the answer.
What is rational equation?
A rational equation is when its solution can be expressed as a fraction
When solving a rational equation why is it necessary to perform a check?
It is important to check your answers to make sure that it doesn't give a zero denominator in the original equation. When we multiply both sides of an equation by the LCM the result might have solutions that are not solutions of the original equation. We have to check possible solutions in the original equation to make sure that the denominator does not equal zero. There is also the possibility that calculation errors were made in solving.
What is the solution of rational equations reducible to quadratic equation?
A quadratic equation ax2 + bx + c = 0 has the solutions x = [-b +/- sqrt(b2 - 4*a*c)]/(2*a)
How is a extraneous solution of a ration equation similar to a excluded value of a rational equation?
An extraneous solution of a rational equation is a solution that arises from the algebraic process of solving the equation but does not satisfy the original equation. This can occur when both sides are manipulated in ways that introduce solutions not valid in the original context. An excluded value, on the other hand, refers to specific values of the variable that make the denominator zero, rendering the equation undefined. Both concepts highlight the importance of checking solutions against the original equation to ensure they are valid.
What statement must be true of an equation before you can use the quadratic formula to find the solutions?
The quadratic formula can be used to find the solutions of a quadratic equation - not a linear or cubic, or non-polynomial equation. The quadratic formula will always provide the solutions to a quadratic equation - whether the solutions are rational, real or complex numbers.
When you divide a rational equation by another rational equation you end up multiplying by its?
Reciprocal. Except that dividing by a rational equation is much easier.
