Finding all rational roots, or solutions, to an equation is an important step in solving an equation. An equation is said to be solved when all possible solutions are found.
Solving quadratic equations can be done by factoring the equation, finding its zero value(s), and then finding all values that satisfy the positive and negative versions of the zero value.
Quadratic equations can also have linear roots, or zeros that are just 0. When this happens, the solution has been solved!
The problem with solving quadratic equations with rational roots is that there can be many of them. There is no way to know how many linear roots (zoes) an equation will have until it is solved.
This article will explain how to find all of the rational roots of a quadratic equation.
Extract the square roots
Once you have the quadratic equation, you can extract its square roots. This is done by putting the quadratic equation in radical form.
To do this, first factor the denominator into its monomial factors. Then, separate out the linear term and put it under the radical. Finally, bring down the next highest power of x and set it equal to one.
For example, we will solve the following quadratic equation: x^4+8x^3+7x^2-40x-60=0. First, we will factor the denominator into its monomials: (x+1)(x^2+2). Then, we will separate out the linear term and put it under the radical: (x+1)(x^2+2) = (x+1)R(x^2+2)=0.
Divide by the square roots
Once you have factored the polynomial, you can find the rational roots by dividing each term in the factor by its corresponding coefficient.
For example, if you had x^4+8x^3+7x^2-40x-60=0, you would divide each term in the factor by x^2, x^3, and x^4, respectively.
The first step is to divide each term in the factor by x^2. When you do this, you get (x+1) as a quotient. Since there is no coefficient for x+1, this is your first rational root!
The next step is to divide each term in the factor by x^3. When you do this, you get (x−1) as a quotient. Since there is no coefficient for x−1, this is your second rational root!
The last step is to divide each term in the factor by x^4. When you do this, you get 1 as a quotient.
Extract the fourth roots
Now that all the complex numbers under 1 have been factored out, you can now find the fourth roots of each. These are 1, i, -1, and -i.
The same process is used to find the fourth roots of a number as to find any root. Start by finding two numbers that when multiplied together are 1. These are called factors of 1. Then divide the number being rooted by one of these factors until you get a non-zero number. This is your root!
For example, to find the fourth root of 8, first find two numbers that when multiplied together are 8. The only ones that fit this description are 2 and 4, so 2 is a factor of 8. Now divide 8 by 2 until you get a non-zero number: 64! So the fourth root of 8 is 64.
Divide by the fourth roots
Once you have found the quadratic equation, the next step is to divide by the fourth roots of each coefficient. This means you would divide by -2, -2, -2, and -2 respectively.
This process is done by using the quadratic formula once more. You would get imaginary numbers as solutions this time around, however.
When you divide both sides of the original equation by x^4, you would get (x/x)^(1/4)=0, which is not true. Imaginary numbers do not satisfy equality statements unless they are 0.
By dividing both sides of the original equation by x^4+8x^3+7x^2-40x-60 instead, you would get (x/x)^(1/4)=0, which is true. This is because all of the roots are 0 when x=0.
Simplify the fractions
Once you have factored the polynomial, you can simplify the fractions that are in the formula. For example, in our equation 8x^2+16x+8=0, we can divide both sides by 8 to get x=0, thus removing the fraction.
When doing this, make sure you check if there are any solutions to the equation that have a fraction. If so, then you need to either find a way to remove the fraction or prove there are no solutions with fractions.
For example, we had to prove there were no solutions to 8x^2+16x+8=0 that had a fraction since we eliminated them when dividing both sides by 8.
Finish writing out the equation
Once you have written out the equation, the next step is to finish writing out the equation. This means adding zeroes to the end of the variable or coefficient part of the equation.
In this case, you would add a zero to the left side of the equation. Once you do this, it will look like X=0, which is not very useful.
So, next, add a zero to the right side of the variable part of the equation. Now it looks like X=0+N, where N is some number that we are trying to find. This is much more useful!
Finally, test out whether or not N=0-if it does, then you are done! If it does not, then repeat this process until N=0.
Find all rational roots for a given equation
So, you have a quadratic equation, and you have found the quadratic formula answer. Now, let’s find all of the rational roots (whole numbers that when multiplied by themselves or each other are equal to the coefficient of the corresponding factor) for this answer.
To find all of the rational roots, we need to first list all of the possible factors of the coefficient of each factor. Then, we need to multiply each of these by every possible whole number exponent and see if they match with our original equation.
For example, let’s use 2 as our first root: 2×2=4; 4+8=12; 12 is not in our original solution set but is a rational root; therefore, 2 is a rational root for our original solution set.
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