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Using the quadratic equation formula or completing the square
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How many ways are there to solve a quadratic equation?
There are so far 8 common methods to solve quadratic equations:GraphingFactoring FOIL methodCompleting the square.Using the quadratic formula (derived from algebraic manipulation of "completing the square" method).The Diagonal Sum Method. It quickly and directly gives the 2 real roots in the form of 2 fractions. In fact, it can be considered as a shortcut of the factoring method. It uses the Rule of Signs for Real Roots in its solving process. When a= 1, it can give the 2 real roots quickly without factoring. Example. Solve x^2 - 39x + 108 = 0. The Rule of Signs indicates the 2 real roots are both positive. Write the factor-sets of c = 108. They are: (1, 108), (2, 54), (3, 36)...Stop! This sum is 36 + 3 = 39 = -b. The 2 real roots are 3 and 36. No needs for factoring! When a is not one, this new method selects all probable root-pairs, in the form of 2 fractions. Then it applies a very simple formula to see which root-pair is the answer. Usually, it requires less than 3 trials. If this new method fails, then this given quadratic equation can not be factored, and consequently the quadratic formula must be used. Please see book titled:"New methods for solving quadratic equations and inequalities" (Amazon e-book 2010).The Bluma MethodThe factoring AC Method (Youtube). This method is considerably improved by a "new and improved AC Method", recently introduced on Google or Yahoo Search.The new Transforming Method, recently introduced, that is may be the best and fastest method to solve quadratic equations. Its strong points are: simple, fast, systematic, no guessing, no factoring by grouping, and no solving the binomials. To know this new method, read the articles titled:"Solving quadratic equations by the new Transforming Method" on Google or Yahoo Search.BEST METHODS TO SOLVE QUADRATIC EQUATIONS. A. When the equation can't be factored, the best choice would be the quadratic formula. How to know if the equation can't be factored? There are 2 ways:1. Start solving by the new Transforming Method in composing factor pairs of a*c (or c). If you can't find the pair whose sum equals to (-b), or b, then the equation can't be factored.2. Calculate the Discriminant D = b^2 - 4ac. If D isn't a perfect square, then the equation can't be factored.B. When the equation can be factored, the new Transforming Method would be the best choice.
What is a discriminant in a quadratic equation?
General form of a quadratic equation is: ax2+b+c = 0 The discriminant is: b2-4ac If the discriminant equals zero then there are two equal roots If the discriminant is greater than zero then there are two different roots If the discriminant is less than zero then there are no real roots
How do you find the point that two equations intersect?
If the equations are in y= form, set the two equations equal to each other. Then solve for x. The x value that you get is the x coordinate of the intersection point. To find the y coordinate of the intersection point, plug the x you just got into either equation and simplify so that y= some number. There are other methods of solving a system of equations: matrices, substitution, elimination, and graphing, but the above method is my favorite!
What is the equation for vertex form?
The vertex form for a quadratic equation is y=a(x-h)^2+k.
How many sides does a polygon have if it has 902 diagonals?
Let n be the number of sides: 1/2*(n2-3n) = diagonals 1/2*(n2-3n) = 902 Multiply both sides by 2 and form a quadratic equation: n2-3n-1804 = 0 Solving the above by means of the quadratic equation formula gives a positive value for n as 44 Therefore the polygon has 44 sides
When solving a quadratic equation by factoring what method is used?
Start with a quadratic equation in the form � � 2 � � � = 0 ax 2 +bx+c=0, where � a, � b, and � c are constants, and � a is not equal to zero ( � ≠ 0 a =0).
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 would you go about solving a quadratic equation?
Write the quadratic equation in the form ax2 + bx + c = 0 then the roots (solutions) of the equation are: [-b ± √(b2 - 4*a*c)]/(2*a)
Where the given equations are not linear?
Equations are not linear when they are quadratic equations which are graphed in the form of a parabola
What are the steps to solving a quadratic equation?
In general, there are two steps in solving a given quadratic equation in standard form ax^2 + bx + c = 0. If a = 1, the process is much simpler. The first step is making sure that the equation can be factored? How? In general, it is hard to know in advance if a quadratic equation is factorable. I suggest that you use first the new Diagonal Sum Method to solve the equation. It is fast and convenient and can directly give the 2 roots in the form of 2 fractions. without having to factor the equation. If this method fails, then you can conclude that the equation is not factorable, and consequently, the quadratic formula must be used. See book titled:" New methods for solving quadratic equations and inequalities" (Trafford Publishing 2009) The second step is solving the equation by the quadratic formula. This book also introduces a new improved quadratic formula, that is easier to remember by relating the formula to the x-intercepts with the parabola graph of the quadratic function.
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.
Why are Quadratic equations which are expressed in the form of ax2 plus bx plus c 0 where a does not equal 0 may have how many solutions?
Why are Quadratic equations, which are expressed in the form of ax2 + bx + c = 0, where a does not equal 0,
What is the discriminant and how does it help in knowing how many solutions a quadratic equations will have?
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. ■
What is the relationship between the roots of quadratic?
Suppose the roots a quadratic, in the form ax2 + bx + c = 0, are p and q. Then p + q = -b/a and pq = c/a
When the roots are equal of a quadratic equation?
Write the quadratic equation in the form ax2 + bx + c = 0 The roots are equal if and only if b2 - 4ac = 0. The expression, b2-4ac is called the [quadratic] discriminant.
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.
Formula of quadratic equation if roots are given?
A quadratic equation has the form: x^2 - (sum of the roots)x + product of the roots = 0 or, x^2 - (r1 + r2)x + (r1)(r2) = 0
