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Answered 2015-12-18 15:07:51
The discriminant must be a perfect square or a square of a rational number.
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When the discriminant is perfect square the answer to a quadratic equation will be?
Rational.
If the discriminant of an equation is negative?
It has two complex solutions.
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.
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 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.
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 the solution of rational equations reducible to quadratic?
They are the solutions for the reduced quadratic.
What is the discriminant?
If a quadratic equation is ax2+bx+cthen we can learn something about the roots withoutcompletely solving the quadratic formula.The discriminant is b2-4ac. You may recognize this as part of the quadratic formula.If the value is a non-zero perfect square, there are 2 rational rootsIf the value is an imperfect square, there are 2 irrational rootsIf the value is zero, there is 1 rational root (parabola vertex is on the x-axis)If the value is negative, there are imaginary roots (no intersection with x-axis)The discriminant, therefore, tells us the nature of the roots.
X2-14x plus 46 equals 0?
This is the generalized trinomial equation (aka quadratic):y = ax2 - bx - cBefore factoring, always check the discriminant of the quadratic equation, which is:b2 - 4acIf it is a rational square (16, 25, 196, 225), then it is factorable. If it is not, then it is not factorable.In this case, it is not, since the discriminant is equal to 2√3.Now, you will have to use the quadratic formula:(-b2 +/- √(b2 - 4ac))/2This will give you (14 +/- 2√3)/2
What is the discriminant in a math equation?
In a quadratic equation of the form ax2+bx + c = 0, the discriminant is b2-4ac. It determines the nature of the roots of the equation. If it is positive, there are two real roots; if is negative, there are two complex roots; if it is zero, there is one real root, often called a double root. Both real roots are rational if and only the discriminant is a perfect square.
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.
Do all rational equations have a single solutions?
No. An equation as simple as x2 = 1 has two solutions.
Which of the folllowing indicates that the roots of a quadratic are irrational?
The discriminant is the expression inside the square root of the quadratic formula. For a quadratic ax² + bx + c = 0, the quadratic formula is x = (-b +- Sqrt(b² - 4ac))/(2a). The expression (b² - 4ac) is the discriminant. This can tell a lot about the type of roots. First, if the discriminant is a negative number, then it will have two complex roots. Because you have a real number plus sqrt(negative) and real number minus sqrt(negative). You asked about irrational. If the discrimiant is a perfect square number {like 1, 4, 9, 16, etc.} then the quadratic will have two distinct rational roots (which are real numbers). If the discriminant is zero, then you will have a double root, which is a real rational number. So if the discrimiant is positive, but not a perfect square, then the roots will be irrational real numbers. If the discriminant is a negative number which is not the negative of a perfect square, then imaginary portion of the complex number will be irrational.
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 do you work out 3x squared -10x equals -5?
This is a quadratic expression in x. It cannot be solved because it is an expression: there is no equation to solve. Furthermore, the discriminant for the quadratic = (-10)2 - 4*3*(-5) = 100+60 = 160 which is not a square number. There are, therefore no rational factors. While it would be possible to give irrational factors, factorising a quadratic into linear terms containing irrational numbers is rarely useful.
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 rational equation?
A rational equation is when its solution can be expressed as a fraction
Discriminant for 4x squared plus 4x plus 1?
B squared -4ac is the discriminant. you get 0, which means it will have one rational solution.
What are some ways to solve a quadratic equation?
Factorisation - only works if the solutions are rational, so not always possible).Graph it - only works if solutions are real. Although this method WILL find irrational soultions, it may be difficult to relate answer to simple surd forms eg very few people could recognise 1.32288.. as sqrt(7)/2!. Also, it is difficult to read solutions off a graph to a high degree of precision.Numerical methods - only works if solutions are real. Can go to high precision but you still have the problem of relating an infinite decimal to a simple surd representation.Use quadratic equation - always provides a solution if you allow complex numbers.
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.
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.
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.
How do you find solutions for cubic and quartic equations?
Although there is a method for cubics, there are no simple analytical ways.Sometimes you may be able to use the remainder theorem to find one solutions. THen you can divide the original equation using that solution so that you are now searching for an equation of a lower order. If you started off with a cubic you will now have a quadratic and, if all else fails, you can use the quadratic formula.You could use a graphic method. A cubic musthave a solution although that solution need not be rational. A quartic need no have any.Lastly, you could use a numeric method, such as the Newton-Raphson iteration.
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.
Is a rational exponent equation?
No, it is an expression, not an equation.
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