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  Solving Equations Algebraically

Solving Equations Algebraically:

Quadratic Equations:

Quadratic equation is an equation which can be written as Ax2 + Bx + C = 0 where A ≠ 0. This form is termed as the standard form.

There are mainly four ways to solve a quadratic equation.

Factoring:

It works fine when the quadratic can simply be factored.

We can know the trial and error method of factoring. The other is the AC Method of Factoring.

The basic idea behind factoring is to put the equation to standard form, and then the factor of left hand side to two factors (x - a) and (x - b). The solutions to equation are then x = a and x = b. The factors will of course differ if A ≠ 1. Factoring works as there is a rule that states when the product of two factors is zero, then one of the factors should be zero.

Extraction of Roots:

It works fine when there is no linear term, that is, whenever B=0.

The extraction of roots is termed as the square root principle. The main goal here is to obtain the squared variable term by itself on one side and a non-negative constant on other side.

Square root of both the sides is then taken. Keep in mind that the square root of x2 is the absolute value of x. Whenever you solve an equation including an absolute value, you will obtain a plus and minus in the solution. Too frequently, we bypass the step with absolute value in it and go straight to the plus or minus phase. This is okay, as long as we keep in mind the reason.

Completing the Square:

It works fine whenever the leading coefficient of A is 1 and B is even.

a) When A is not 1, then either divide every term by A and hence it is 1 or factor an A out of the variable terms only (that is, not out of the constant).

b) Move the constant to right hand side. Be sure and leave the space at end of left hand side before the equal sign for the constant which will be inserted in there afterward.

c) Take 1/2 of linear coefficient (B) and call that number ‘b’. In next line, write (x + b)2 =. We will later fill it in the right hand side. Obviously, if B is negative, then the expression will look similar to (x - b)2 =.

d) Square that value you simply found (one-half of B) and write it in that place you left at the end of left hand side before the equal sign on the preceding line. Add similar value to the right hand side, too. It is very significant that we add similar thing to both sides. If you choose to factor out A, instead of dividing via by it, make sure that you add up A times that constant to the right hand side.

e) Simplify the right hand side.

f) Carry on the procedure as an extraction of roots problem.

Quadratic Formula:

The quadratic formula is a catch-all which can be employed to solve any quadratic equation. The equation should first of all be written in the standard form, and then coefficients plugged to the formula. Formula was derived in the class by completing the square on the generic quadratic equation.

If ax2 + bx + c = 0 and a ≠ 0, then

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When the solutions from quadratic formula are rational (that is, no radicals), then the equation could have been resolved by factoring.

Discriminant b2 - 4ac:

The discriminant is the radicand from radical in quadratic formula. Depending on what kind of number the discriminant is, we can state what kind and how many solutions there will be.

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Polynomials of Higher Degree:

We can try factoring. Be sure to confirm for the greatest common factor first.

Equations involving Radicals:

a) Isolate the radical term on one side.
b) Squaring both sides of equation.
c) Solve for x.
d) Check your outcomes or answers. There might be extraneous solutions.

Equations with more than one radical term:

a) Isolate the one radical term on one side. It does not matter which one. Don’t square both sides of the equation when both the radicals are on similar side of equal sign. This will work out when you are cautious, although it will be very messy.

b) Square both the sides of equation.

c) Simplify.

d) Isolate the enduring radical on a side by itself. Keep in mind that there will be a radical since of the middle term whenever you square a binomial.

e) Squaring both sides again.

f) Solve for x.

g) Check your outcomes or answers are really significant here. You applied a non 1-1 function twice and thus there really could be extraneous solutions. Be sure to verify.

Equations involving Fractions:

Fractions are no doubt your friends. Whenever you need to give a solution, they are favored over decimals. If you have the choice as to working with the decimals or working with fractions, always select the fraction if the decimals don’t terminate. Though, if you have the option, and you do with equations, of not working with either one - get it!

a) Find out the LCD (least common denominator).

b) Set the LCD = 0 and note what values of x can’t be used. We will be removing the denominators in the next step and the domain will no longer be implied, thus it is essential to define the restrictions.

c) Multiply each and every term of both sides by LCD and simplify.

d) Check your outcome or answer against your limitations to make sure you are not using an extraneous solution.

Equations involving Absolute Value:

There are mainly two possible values that have similar absolute value. Keep in mind, that the absolute value is a piece-wisely stated function. Thus, whenever solving an equation having an absolute value you should create two equations, one for each and every piece.

Also note the limitations whenever you break the equation up to its two parts. This is possible to obtain extraneous solutions. If you do not want to take the time to keep track of the limitations, then

•    Do not be surprised whenever you miss the problem on the exam or
•    Check all of your solutions back to the original equation.

Position Equation:

The height, s, in feet of a freely falling body (close to the surface of earth), after t seconds, can be modeled by the function as:

s(t) = -16 t2 + v0t + s0
v0 = initial velocity and s0 = initial height

-16 is valid for the earth's surface and will modify depending on the celestial body exerting the gravitational pull. For those concerned, the quadratic coefficient will always be one-half the acceleration due to the gravity (-32 ft/s/s on the earth). Rest of the equation is valid no matter what body you are on.

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