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What is the discriminant and how does it help in knowing how many solutions a quadratic equations will have?
Wiki User
∙ 11y ago
Put the equation into ax²+bx+c=0 form. The discriminant is b²-4ac. If it is negative, there are no real roots. If it is 0, there is one real root. If it is positive, there are 2 real roots. ■
Wiki User
∙ 11y ago
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What are the solutions to the equation 2x2 plus 3x-50?
Without an equality sign the given quadratic expression can't be classed as an equation but knowing how to use the quadratic equation formula would be helpful when given such problems.
How many times does the graph of the function below touch or cross the x-axis 4x2-7x 40?
Without knowing the plus or minus value of 40 it's difficult to say but in general:- If the discriminant of a quadratic equation = 0 then it touches the x axis at 1 point If the discriminant is greater than zero then it touches the x axis at 2 points If the discriminant is less than zero then it does not touch the x axis
Can all quadratic equations be solved?
Well, that depends on what you mean "solve by factoring." For any quadratic equation, it is possible to factor the quadratic, and then the roots can be recovered from the factors. So in the very weak sense that every quadratic can be solved by a method that involves getting the factors and recovering the roots from them, all quadratic equations can be solved by factoring. However, in most cases, the only way of factoring the quadratic in the first place is to first find out what its roots are, and then use the roots to factor the quadratic (any quadratic polynomial can be factored as k(x - r)(x - s), where k is the leading coefficient of the polynomial and r and s are its two roots), in which case trying to recover the roots from the factors is redundant (since you had to know what the roots were to get the factors in the first place). So to really count as solving by factoring, it makes sense to require that the solution method obtains the factors by means that _don't_ require already knowing the roots of the polynomial. And in this sense, most quadratic equations are not solvable through factoring.
How many solutions are there to the systems of equations?
I'm not 100% certain about what you're asking, but each function and relation can have different solutions, either one of the three ways. (Always dealing with two equations) This is based off my Grade 10 Knowledge One (Intersecting at one specific point) None (Parallel Lines) Coincident two lines with the same slope and intercept) Refer back to : " y=mx+b " equation if needed. If you are talking about the possible ways to find the solution (x,y), there are also three. Elimination, (Removing one variable to solve the equation) Substitution, (Knowing what x or y, and inputting in the second equation) Graphing, (By drawing both equations, This method is not very accurate)
How many existing methods are there in solving quadratic equations?
There are 5 existing methods in solving quadratic equations. For the first 4 methods (quadratic formula, factoring, graphing, completing the square) you can easily find them in algebra books. I would like to explain here the new one, the Diagonal Sum Method, recently presented in book titled:"New methods for solving quadratic equations and inequalities" (Trafford 2009). It directly gives the 2 roots in the form of 2 fractions, without having to factor the equation. The innovative concept of the method is finding 2 fractions knowing their Sum (-b/a) and their Product (c/a). It is very fast, convenient and is applicable whenever the given quadratic equation is factorable. In general, it is hard to tell in advance if a given quadratic equation can be factored. However, if this new method fails to find the answer, then we can conclude that the equation can not be factored, and consequently, the quadratic formula must be used. This new method can replace the trial-and-error factoring method since it is faster, more convenient, with fewer permutations and fewer trials.
What are the solutions to the equation 2x2 plus 3x-50?
Without an equality sign the given quadratic expression can't be classed as an equation but knowing how to use the quadratic equation formula would be helpful when given such problems.
What are the solutions of the quadratic equation 3x2 - x 11?
Without an equality sign and not knowing the plus or minus value of 11 it can't be considered to be an equation.
How many times does the graph of the function below touch or cross the x-axis 4x2-7x 40?
Without knowing the plus or minus value of 40 it's difficult to say but in general:- If the discriminant of a quadratic equation = 0 then it touches the x axis at 1 point If the discriminant is greater than zero then it touches the x axis at 2 points If the discriminant is less than zero then it does not touch the x axis
Which the easy way the method of factoring or the solving the quadratic equation?
By knowing how to use the quadratic equation formula.
Can all quadratic equations be solved?
Well, that depends on what you mean "solve by factoring." For any quadratic equation, it is possible to factor the quadratic, and then the roots can be recovered from the factors. So in the very weak sense that every quadratic can be solved by a method that involves getting the factors and recovering the roots from them, all quadratic equations can be solved by factoring. However, in most cases, the only way of factoring the quadratic in the first place is to first find out what its roots are, and then use the roots to factor the quadratic (any quadratic polynomial can be factored as k(x - r)(x - s), where k is the leading coefficient of the polynomial and r and s are its two roots), in which case trying to recover the roots from the factors is redundant (since you had to know what the roots were to get the factors in the first place). So to really count as solving by factoring, it makes sense to require that the solution method obtains the factors by means that _don't_ require already knowing the roots of the polynomial. And in this sense, most quadratic equations are not solvable through factoring.
What is the special cases of quadratic equation?
The standard form of a quadratic equation is: ax^2 + bx + c = 0. Depending on the values of the constants (a, b, and c), a quadratic equation may have 2 real roots, one double roots, or no real roots.There are many "special cases" of quadratic equations.1. When a = 1, the equation is in the form: x^2 + bx + c = 0. Solving it becomes solving a popular puzzle: find 2 numbers knowing their sum (-b) and their product (c). If you use the new Diagonal Sum Method (Amazon e-book 2010), solving is fast and simple.Example: Solve x^2 + 33x - 108 = 0.Solution. Roots have opposite signs. Write factor pairs of c = -108. They are: (-1, 108),(-2, 54),(-3, 36)...This sum is -3 + 36 = 33 = -b. The 2 real roots are -3 and 36. There is no needs for factoring.2. Tips for solving 2 special cases of quadratic equations.a. When a + b + c = 0, one real root is (1) and the other is (c/a).Example: the equation 5x^2 - 7x + 2 = 0 has 2 real roots: 1 and 2/5b. When a - b + c = 0, one real roots is (-1) and the other is (-c/a)Example: the equation 6x^2 - 3x - 9 = 0 has 2 real roots: (-1) and (9/6).3. Quadratic equations that can be factored.The standard form of a quadratic equation is ax^2 + bx + c = 0. When the Discriminant D = b^2 - 4ac is a perfect square, this equation can be factored into 2 binomials in x: (mx + n)(px + q)= 0. Solving the quadratic equation results in solving these 2 binomials for x. Students should master how to use this factoring method instead of boringly using the quadratic formula.When a given quadratic equation can be factored, there are 2 best solving methods to choose:a. The "factoring ac method" (You Tube) that determines the values of the constants m, n, p, and q of the 2 above mentioned binomials in x.b. The Diagonal Sum Method (Amazon ebook 2010) that directly obtains the 2 real roots without factoring. It is also considered as "The c/a method", or the shortcut of the factoring method. See the article titled" Solving quadratic equations by the Diagonal Sum Method" on this website.4. Quadratic equations that have 2 roots in the form of 2 complex numbers.When the Discriminant D = b^2 - 4ac < 0, there are 2 roots in the form of 2 complex numbers.5. Some special forms of quadratic equations:- quadratic equations with parameters: x^2 + mx - 7 + 0 (m is a parameter)- bi-quadratic equations: x^4 - 5x^2 + 4 = 0- equations with rational expression: (ax + b)/(cx + d) = (ex + f)- equations with radical expressions.
How many solutions are there to the systems of equations?
I'm not 100% certain about what you're asking, but each function and relation can have different solutions, either one of the three ways. (Always dealing with two equations) This is based off my Grade 10 Knowledge One (Intersecting at one specific point) None (Parallel Lines) Coincident two lines with the same slope and intercept) Refer back to : " y=mx+b " equation if needed. If you are talking about the possible ways to find the solution (x,y), there are also three. Elimination, (Removing one variable to solve the equation) Substitution, (Knowing what x or y, and inputting in the second equation) Graphing, (By drawing both equations, This method is not very accurate)
How many existing methods are there in solving quadratic equations?
There are 5 existing methods in solving quadratic equations. For the first 4 methods (quadratic formula, factoring, graphing, completing the square) you can easily find them in algebra books. I would like to explain here the new one, the Diagonal Sum Method, recently presented in book titled:"New methods for solving quadratic equations and inequalities" (Trafford 2009). It directly gives the 2 roots in the form of 2 fractions, without having to factor the equation. The innovative concept of the method is finding 2 fractions knowing their Sum (-b/a) and their Product (c/a). It is very fast, convenient and is applicable whenever the given quadratic equation is factorable. In general, it is hard to tell in advance if a given quadratic equation can be factored. However, if this new method fails to find the answer, then we can conclude that the equation can not be factored, and consequently, the quadratic formula must be used. This new method can replace the trial-and-error factoring method since it is faster, more convenient, with fewer permutations and fewer trials.
How do you solve quadratic equations by factoring?
There is a new method, called Diagonal Sum Method, that quickly and directly give the 2 roots without having to factor the equation. The innovative concept of this method is finding 2 fractions knowing their sum (-b/a) and their product (c/a). It is fast, convenient and is applicable to any quadratic equation in standard form ax^2 + bx + c = 0, whenever it can be factored. If it fails to find answer, then the equation is not factorable, and consequently, the quadratic formula must be used. So, I advise you to proceed solving any quadratic equation in 2 steps. First, find out if the equation can be factored? How?. Use this new method to solve it. It usually takes fewer than 3 trials. If its fails then use the quadratic formula to solve it in the second step. See book titled:" New methods for solving quadratic equations and inequalities" (Trafford Publishing 2009)
What is the x-coordinate of the point where the lines x4y8 and 12x y11 intersect?
Without any equality signs and not knowing the plus or minus values of the given terms they can't be considered to be straight line equations and so therefore no solutions are possible.
What are two algebraic methods for solving quadratic equations?
Finally, there are two methods to use, depending on if the given quadratic equation can be factored or not. 1.- The first one is the new Diagonal Sum Method, recently presented in book titled: "New methods for solving quadratic equations" (Trafford 2009). This method directly gives the two roots in the form of two fractions, without having to factor it. The innovative concept of this new method is finding 2 fractions knowing their product (c/a) and their sum (-b/a). This new method is applicable to any quadratic equation that can be factored. It can replace the existing trial-and-error factoring method since this last one contains too many more permutations. In general, it is hard to tell in advance if a given quadratic equation can be factored. However, if the new method fails to get the answers, then you can positively conclude that this equation can not be factored. Consequently, the quadratic formula must be used in solving. We advise students to always try to solve the given equation by the new method first. If the student gets conversant with this method, it usually take less than 2 trials to get answers. 2. the second one uses the quadratic formula that students can find in any algebra book. This formula must be used for all quadratic equations that can not be factored.
How are some of the graphs of cubic equations different?
The question cannot be answered without knowing what they are meant to be different from!
