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Why does the quadratic formula work every time for quadratic equations?
Wiki User
∙ 13y ago
The quadratic formula works for every quadratic equation because it is the standard form of a quadratic solved for x.
ax2+bx+c=0 - Standard Form, how quadratic equations are normally displayed
x2+bx/a+c/a=0 - Divide both sides by a, Division Property of =
x2+(b/a)x= -(c/a) - Separate variables and constants, Subtraction Property of =
x2+(b/a)x+(b/2a)2=(b/2a)2-(c/a) - Complete the square. (b/2)2
(x+b/2a)2=(b2-4ac)/4a2 - Factor and Simplify
x+b/2a=+/-sqrt((b2-4ac)/4a2) - Square Root both sides.
x=-(b/2a)+/-sqrt(b2-4ac)/2a - Solve for x, Subtraction Property of =
x=(-b+/-sqrt(b2-4ac))/2a - The Quadratic Equation, Simplify
Wiki User
∙ 13y ago
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Advantages and disadvantages of using journals in mathematics?
Somebody (possibly in seventh-century India) was solving a lot of quadratic equations by completing the square. At some point, he noticed that he was always doing the exact same steps in the exact same order for every equation. Taking advantage of the one of the great powers and benefits of algebra (namely, the ability to deal with abstractions, rather than having to muck about with the numbers every single time), he made a formula out of what he'd been doing:The Quadratic Formula: For ax2 + bx + c = 0, the value of x is given byThe nice thing about the Quadratic Formula is that the Quadratic Formula always works. There are some quadratics (most of them, actually) that you can't solve by factoring. But the Quadratic Formula will always spit out an answer, whether the quadratic was factorable or not.I have a lesson on the Quadratic Formula, which gives examples and shows the connection between the discriminant (the stuff inside the square root), the number and type of solutions of the quadratic equation, and the graph of the related parabola. So I'll just do one example here. If you need further instruction, study the lesson at the above hyperlink.Let's try that last problem from the previous section again, but this time we'll use the Quadratic Formula:Use the Quadratic Formula to solve x2 - 4x - 8 = 0.Looking at the coefficients, I see that a = 1, b = -4, and c = -8. I'll plug them into the Formula, and simplify. I should get the same answer as before:
What is the use of quadratic equation in your real life?
Chemist use the quadratic equation all the time to find concentrations in equilibrium reactions.Mechanics in physics, kinematics to be precise, use this formula to find position, time velocity or acceleration of moving objects.X = Vot + 1/2at2--------------------------------a quadratic formula
Can X equal a negative radical using the quadratic formula?
Yes any time 4ac > b2 then x will be a complex numberthe curve represented by that formula will have no x intercepts.Yes.
In what cases might understanding Quadratic Functions and Equations be important?
Quadratic Equations are used every day, like in most sports or anything you may throw because it kind of reminds people of the gravitational pull. In my opinion where you would see it a great deal is volley ball. Every time you set, serve or bump the ball it is going to go up and come down or if u mess up it is going to go down and then bounce back up. So as it could be imagined a volley ball does have a maximum and minimum height depending on how it is hit. A volley ball will go in the general shape of a parabola. -Asha Omar
Why is it advantageous to know a variety of ways to solve and graph quadratic equations?
since there'll always be a time when one or two don't work. also, it's easier to check work by using another strategy
What is the quadratic formula used for?
The quadratic formula is used all the time to solve quadratic equations, often when the factors are fractions or decimals but sometimes as the first choice of solving method. The quadratic formula is sometimes faster than completing the square or any other factoring methods. Quadratic formula find: -x-intercept -where the parabola cross the x-axis -roots -solutions
What are the applications of quadratic equations in every day life?
Police, Quadratics, Action! If you know the initial speed of car, how far you are travelling and what your acceleration is, there is a special formula that lets you find out how long the journey will take. This formula is a quadratic with time as its unknown quadratic quantity. The police use this equation - along with many other quadratic and non-quadratic equations - when they attend a road traffic accident (RTA). They do this to find out if the driver was breaking the speed limit or driving without due care and attention. They can discover how fast the car was going at the time the driver started braking and how long they were braking for before they had the accident. This is done by finding the road's coefficient of friction and by measuring the length of the skid marks of the vehicles involved. Once they have this information they turn to Mathematics and the trusted quadratic equation. Einstein's Famous Quadratic The most famous equation in the world is technically quadratic. Einstein discovered the formula: Where E is the Energy of an object, m is its mass and c is the speed of light. This formula relates mass and energy and came from Einstein's work on Special and General Relativity. However, in practice it is not solved as a quadratic equation as we know the value of the speed of light. For more information on Einstein and his Theory of Special Relativity see the links at the bottom of the page. There are many more uses for quadratic equations. For more information please see the links to "101 Uses of a Quadratic Equation" at the bottom of the page.
Advantages and disadvantages of using journals in mathematics?
Somebody (possibly in seventh-century India) was solving a lot of quadratic equations by completing the square. At some point, he noticed that he was always doing the exact same steps in the exact same order for every equation. Taking advantage of the one of the great powers and benefits of algebra (namely, the ability to deal with abstractions, rather than having to muck about with the numbers every single time), he made a formula out of what he'd been doing:The Quadratic Formula: For ax2 + bx + c = 0, the value of x is given byThe nice thing about the Quadratic Formula is that the Quadratic Formula always works. There are some quadratics (most of them, actually) that you can't solve by factoring. But the Quadratic Formula will always spit out an answer, whether the quadratic was factorable or not.I have a lesson on the Quadratic Formula, which gives examples and shows the connection between the discriminant (the stuff inside the square root), the number and type of solutions of the quadratic equation, and the graph of the related parabola. So I'll just do one example here. If you need further instruction, study the lesson at the above hyperlink.Let's try that last problem from the previous section again, but this time we'll use the Quadratic Formula:Use the Quadratic Formula to solve x2 - 4x - 8 = 0.Looking at the coefficients, I see that a = 1, b = -4, and c = -8. I'll plug them into the Formula, and simplify. I should get the same answer as before:
What is the use of quadratic equation in your real life?
Chemist use the quadratic equation all the time to find concentrations in equilibrium reactions.Mechanics in physics, kinematics to be precise, use this formula to find position, time velocity or acceleration of moving objects.X = Vot + 1/2at2--------------------------------a quadratic formula
How do students apply quadratic equation as an activity?
Teachers can find many ways to teach students the quadratic equation. An activity could include having contests where students race to solve the equations in the fastest time.
How do engineers use the quadratic formula?
In problems of motion, especially involving constant acceleration, a quadratic equation will from the formulas of motion to solve for time, usually. This is just one example.
What are the pros and cons of solving quadratics by using the quadratic formula?
One pro of using the quadratic formula is that it will produce complex (imaginary) roots just as easily as it can produce real roots. (Factoring with imaginary numbers is a kind of a nightmare!) Another pro to the quadratic formula is that it eliminates the frustrating guess-and-check process. A con of the quadratic formula is that, when it comes to more simple problems, it is usually more time-consuming. A lot of textbook problems are quite easy to factor in your head--it is often not worth the effort of plugging numbers into a long formula. A second con of the quadratic formula is that it is quite long--you might write out the formula, accidentally forget a letter, and whole thing is useless. It's much easier to see that your work is correct when you're factoring.
Can X equal a negative radical using the quadratic formula?
Yes any time 4ac > b2 then x will be a complex numberthe curve represented by that formula will have no x intercepts.Yes.
What you is a formula?
A formula can be the method used in math equations; the ingredients used to make up a baby's bottle at feeding time; or the instructions used to process data in a spreadsheet.
In what cases might understanding Quadratic Functions and Equations be important?
Quadratic Equations are used every day, like in most sports or anything you may throw because it kind of reminds people of the gravitational pull. In my opinion where you would see it a great deal is volley ball. Every time you set, serve or bump the ball it is going to go up and come down or if u mess up it is going to go down and then bounce back up. So as it could be imagined a volley ball does have a maximum and minimum height depending on how it is hit. A volley ball will go in the general shape of a parabola. -Asha Omar
Why is it advantageous to know a variety of ways to solve and graph quadratic equations?
since there'll always be a time when one or two don't work. also, it's easier to check work by using another strategy
How do you solve 4X squared plus 8X minus 20 equals 0?
-- Although not required, you can make it slightly easier on yourselfif you first divide each side of the equation by 4.-- Either way, you're left with a quadratic equation that isn't easilyfactorable by inspection, so you're forced to use the quadratic formulawhich you learned and then forgot.-- The quadratic formula says that when you have the quadratic equation[ Ax2 + Bx + C = 0 ],which is exactly what you have, the two solutions arex = 1/2A [ -B + sqrt(B2-4AC) ]andx = 1/2A [ -B - sqrt(B2-4AC) ].This is the perfect time to use the quadratic formula,and that's exactly how you solve your equation.
