How do I do this ? I'm confused.?

Question

2. What three techniques can be used to solve a quadratic equation? Demonstrate these techniques on the equation "12x^2 - 10x - 42 = 0".

Answer 1

quadratic formula -- x = [-b +/- sqrt (b^2 - 4ac)] / 2a

a = 12, b = -10, c = -42; plug these into the formula

With the "+/-" part, after simplifying the formula break it up into two problems, one with the + and one with the -. There would be two solutions.

factoring -- trial and error

First factor out 2 from the polynomial and divide both sides by 2.

6x^2 - 5x - 21 = 0

Factor the polynomial to two binomials (_ + _)(_ + _). You might use one or two minus signs instead of plus signs.

Use reverse FOIL (first, outer, inner, last).

Break up the first-term coefficient (6) into two factors, putting them as the coefficients of the first terms in the binomials.

Do the same for the last term (-21) and the last terms of the binomials.

Calculate the middle-term coefficient from your binomials, seeing if you get the right number. If not change the numbers being used.

Set each binomial factor equal to 0 and solve (two problems and solutions). This is because for the equation to be true (both sides = 0), either of these binomials has to be 0.

factoring -- completing the square

First you can factor out 2.

2(6x^2 - 5x - 21) = 0

Divide both sides by 2.

6x^2 - 5x - 21 = 0

Multiply the coefficients of the first and last terms (6* -21 = -126) and figure out two numbers that multiply to the result (-126) and add to the coefficient of the middle term (-5).

They are 9 and -14.

Rewrite the polynomial, breading up the middle term into two x-terms with those coefficients.

6x^2 + 9x - 14x - 21 = 0

Break up the polynomial into two parts and factor a common factor out of each part.

3x(2x+3) -7(2x+3) = 0

Factor out common factor 2x+3.

(2x+3)(3x-7) = 0

Either factor has to be 0 for the equation to be true (right side = 0). Write and solve two equations 2x+3=0 and 3x-7=0

There would be two solutions.

Answer 2

a. Completing the square.

b. Factoring.

c. Quadratic formula.