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Why is factoring a valuable tool for solving quadratic equations?
In some simple cases, factoring allows you to find solutions to a quadratic equations easily.Factoring works best when the solutions are integers or simple rational numbers. Factoring is useless if the solutions are irrational or complex numbers. With rational numbers which are relatively complicated (large numerators and denominators) factoring may not offer much of an advantage.
When the discriminant is a perfect square are the solutions rational or irrational?
The "discriminant" here refers to the part of the quadratic equation under the radical (square root) sign. When it is a perfect square, the square root is also a perfect square, so the radical goes away, leaving only rational numbers. So, when the discriminant is a perfect square, the solutions are (usually) rational. Unless, of course, some other part of the result is irrational. For example, if the coefficient of the x2 term ("a" in the quadratic formula) is pi, and the constant term is 1/pi, the discriminant will turn out to be 4 (4ac = 4 * pi * 1/pi = 4), which is a perfect square, but solutions will be irrational anyway because the denominator becomes 2pi, and pi is irrational.
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 happens if there are no zeros in a quadratic function?
Whether or not a function has zeros depends on the domain over which it is defined.For example, the linear equation 2x = 3 has no zeros if the domain is the set of integers (whole numbers) but if you allow rational numbers then x = 1.5 is a zero.A quadratic function such as x^2 = 2 has no rational zeros, but it does have irrational zeros which are sqrt(2) and -sqrt(2).Similarly, a quadratic equation need not have real zeros. It will have zeros if the domain is extended to the complex field.In the coordinate plane, a quadratic without zeros will either be wholly above the horizontal axis or wholly below it.
What is an equation that is not true for any value of the variable?
It is an equation with no solutions [in the given domain]. There may (or may not) be solutions if you change the domain.For example, if X is an integer, then 5X = 2 has no solution. But if you change the domain to rational numbers, then X = 2/5 or 0.4 is a solution.
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 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.
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)
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.
What is the solution of rational equations reducible to quadratic?
They are the solutions for the reduced quadratic.
When the discriminant is perfect square the answer to a quadratic equation will be?
Rational.
How do you know if a quadratic equation can be factored?
The answer depends on what the factors will be. For example, every quadratic can be factored if you allow complex numbers. If not, then it helps to use the discriminant. If it is positive, there are two real factors or solutions. If that positive number is a perfect square, then the factors are rational numbers. If not, they are real but not rational (irrational). If the discriminant is 0, there is one real solution. Lastly, if it is negative, there are no real solutions.
Does there exist a quadratic equation whose coefficients are irrational but both the roots are rational?
None, if the coefficients of the quadratic are in their lowest form.
If the discriminant of an equation is negative?
It has two complex solutions.
Why is factoring a valuable tool for solving quadratic equations?
In some simple cases, factoring allows you to find solutions to a quadratic equations easily.Factoring works best when the solutions are integers or simple rational numbers. Factoring is useless if the solutions are irrational or complex numbers. With rational numbers which are relatively complicated (large numerators and denominators) factoring may not offer much of an advantage.
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.
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.
