discriminant
imaginary
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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.
That's true. Complex and pure-imaginary solutions come in 'conjugate' pairs.
The real solutions are the points at which the graph of the function crosses the x-axis. If the graph never crosses the x-axis, then the solutions are imaginary.
solutions
imaginary
roots
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.
That's true. Complex and pure-imaginary solutions come in 'conjugate' pairs.
The real solutions are the points at which the graph of the function crosses the x-axis. If the graph never crosses the x-axis, then the solutions are imaginary.
When you graph the quadratic equation, you have three possibilities... 1. The graph touches x-axis once. Then that quadratic equation only has one solution and you find it by finding the x-intercept. 2. The graph touches x-axis twice. Then that quadratic equation has two solutions and you also find it by finding the x-intercept 3. The graph doesn't touch the x-axis at all. Then that quadratic equation has no solutions. If you really want to find the solutions, you'll have to go to imaginary solutions, where the solutions include negative square roots.
A quadratic equation is wholly defined by its coefficients. The solutions or roots of the quadratic can, therefore, be determined by a function of these coefficients - and this function called the quadratic formula. Within this function, there is one part that specifically determines the number and types of solutions it is therefore called the discriminant: it discriminates between the different types of solutions.
0 real solutions. There are other solutions in the complex planes (with i, the imaginary number), but there are no real solutions.
If the discriminant of a quadratic equation is less then 0 then it will have no real solutions.
The determinant.The determinant is the part under the square root of the quadratic equation and is:b2-4ac where your quadratic is of the form: ax2+bx+cIf the determinant is less than zero then you have 'no real solutions' (as the square root of a negative number is imaginary.)If the determinant is = 0, then you have one real solution (because you can discount the square root of the quadratic equation)If the determinant is greater than zero you have two real solutions as you have (-b PLUS OR MINUS the square root of the determinant) all over 2aTo find the solutions where they exist you'll need to solve the quadratic formula or use another method.
The term inside the square root symbol is called the radicand. There isn't a specific term for it based on its sign; whether it's positive or negative, it's still the radicand.I'm a little confused by your reference to the quadratic equation.If the radicand is negative, the root is an imaginary number, though that doesn't specifically have anything to do with the quadratic equation in particular.If the quantity b2 - 4ac is negative in the quadratic equation, the root of the quadratic equation is either complex or imaginary depending on whether or not b is zero.---------------------------Thank you to whoever answered this first; you saved me a bit of trouble explaining this to the asker :)However, in the quadractic equation, the number under the radical is called the discriminant. This determines the number of solutions of the quadratic. If the radicand is negative, this means that there are no real solutions to the equation.