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Click here to check out ALL difficulties on Quadratic EquationsQuestion 201614: just how do you know if a quadratic equation will have one, two, or no solutions? how do you find a quadratic equation if you are only given the solution? Is it possible to have various quadratic equations with the very same solution? Explain. Carry out your classmate�s v one or two remedies with which they must create a quadratic equation uncovered 2 services by jsmallt9, solver91311:Answer by jsmallt9(3758) (Show Source): You can put this equipment on her website! 1. How to determine the variety of solutions.Let"s look in ~ the quadratic formula:The an essential is the expression in the square root: . In basic there are three cases: is positive. The square root of a positive number is also some hopeful number. Therefore in the numerator of the quadratic formula we will gain two values: (-b + the square root) and also (-b - the square root). So once we obtain two solutions. Is zero. The square root of zero is zero. For this reason in the numerator we acquire (-b + 0) and (-b - 0). Yet both that these room equal come -b! So when us only obtain one solution. Is negative. And also what is the square source of a an adverse number? What have the right to we square and also get a negative number together an answer? Answer: Nothing. You can not square any type of Real number and also get a negative. So as soon as there are no solutions.2. Recognize the equation native the solution(s)One method to find solutions indigenous the equation is to aspect it. Because that example, solvingwe variable it:For a product to be zero one of the components must it is in zero. In other "words":x-2 = 0 or x-3 =0Solving these us get:x=2 or x=3Now what you want is to have the ability to do this in reverse. Well every the steps above are reversible. Therefore, if we have actually two solutions: and , the equation is going come be:Some examples:Solution: x = 1 or x = 6Equation: (x-1)(x-6) = 0 which gives Solution: x = 10 or x = 0Equation: (x-10)(x-0) = 0 or (x-10)(x) = 0 which provides Solution: x = -3 or x = 1.7Equation: (x-(-3))(x-1.7) = 0 or (x+3)(x-1.7)=0 (You multiply the end this one!?)If you are given that there is only one equipment to a quadratic equation then the equation is of the form: . Because that exmaple, if the only solution come to a quadratic equation is 20, climate the equation would certainly be: which provides .3. Can various quadratic equations have the exact same solution? well it depends on what you median by "different". Yes. Because that example: and all have actually same solution: x-4 or x=5. But are these "different" equations? They sure look different. Yet if you divide both political parties of the 2nd equation by 2 you get the an initial equation. If you division both political parties of the 3rd equation by -5 you get the an initial equation. So room these equations "different"? In mine opinion this equations are not different and that the prize to concern #3 is no. Prize by solver91311(24713) (Show Source): You have the right to put this solution on her website! exactly how do you know if a quadratic equation will have one, two, or no solutions?.The short answer is, it is easy: castle all have two solutions. The an essential Theorem the sdrta.net guarantees it. Currently if what girlfriend really meant to asking is: exactly how do you understand if a quadratic equation will have actually one, two, or no remedies over the real numbers, then check out on.A quadratic equation that the form:has a general solution:The expression under the radical in the general solution, specific is referred to as the discriminant. The discriminant can be evaluated to determine the character of the solutions of a quadratic equation, thus:If , thenif 0">, climate the quadratic has actually two unique real number roots. Furthermore, if is a perfect square number, then the roots will certainly be rational, otherwise the roots of the equation will certainly be a conjugate pair of irrational number of the type whereby if , climate the quadratic has actually a single real number root with a multiplicity of 2. In this case the quadratic is a perfect square having two factors: , hence is the root, and the multiplicity of 2 comes from the reality that there are two the same factors.if , climate the quadratic has actually no actual number solutions. It has actually a conjugate pair of complicated roots the the kind wherein and is the imaginary number characterized by ======================================================How perform you uncover a quadratic equation if you space only given the solution?Note, that have to read "solutions"If the remedies of a quadratic equation are and also , then you deserve to say: and, therefore and finally,Hence deriving the quadratic equation indigenous the solutions. Keep in mind that if you space only given one number as a solution, then you have three possibilities. The very first is that you may have a rational number, in which instance you have two identical factors that end up being a perfect square trinomial: . The second possibility is that you are provided an irrational number, such as v the opportunity that . In this case, the second root is the conjugate of the provided root, specific . The 3rd possibility is the you are given a complicated number root, . Again, the 2nd root is the conjugate: ======================================================Is it possible to have different quadratic equations with the very same solution?Well, that relies on what you median by "different". Because that a provided , , and also , is identical to, and has the very same roots as for every (actually that is true for every yet let"s stick with rational coefficients for your purposes). Because the equations room equivalent, the is a stretch, for me anyway, to call them "different." However, what is true is the the analogous functions, i.e.

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are very different in that they all have different graphs.So while the is true that is various than (see graphs below), the equations and also room equivalent due to the fact that you have the right to multiply the second one through to acquire the first. Note that the graphs crossing the -axis in ~ the same points:======================================================There need to be enough information above for you to come up v a set of services for a quadratic for her classmates, an especially in the answer come the 2nd question.John
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