What is quadratic equation does it always = 0? Solve/Explain?

Question

PROBLEM SOLVE:
x² -6x = 216

I know how to factor it's just that 216 is a big freaking # I can't break it down, and even more harder is when I ever I get a number that matches I need to subtract so I can get "6" IMPOSSSIBLE!

Answer 1

x² - 6x - 216 = 0
x = [ 6 ± √(36 + 864) ] / 2
x = [ 6 ± √900 ] / 2
x = [ 6 ± 30 ] / 2
x = 18 , x = - 12

Answer 2

The equation does need to be restated to equal zero before the factors are found. This is based on the "zero products rule" which says " If a product of several terms = 0 then one or more of the terms = 0". So we set quadratics to = 0 so we can find their roots, or the answers to the problem.

This becomes : x^2 - 6x - 216 = 0

You need to find 2 numbers that multiply to 216 whose DIFFERENCE is 6.
Well 216 is actually an easy number to break down since its factors are 2,2,2,3,3,3 -- and any grouping of these. Just notice that 2 x 2 x 3 = 12 and 3 x 3 x 2 = 18. All 6 factors are accounted for, and 18 - 12 = 6 so the numbers for the parentheses are 12 and 18. The larger must be negative for the sum to be -6, so the numbers needed are therefore -18 and 12.

The factors of this are therefore (x -18)(x + 12).
The numbers that work in the equation (that is, the roots of the above) are 18 and -12
You get this by solving each parenthesis like this :
x - 18 = 0 so x = 18 : add 18 to both sides
x + 12 = 0 so x = -12 : subtract 12 from both sides

Answer 3

I agree with you. It's hard get two numbers which multiply to give 216 and add to give -6. Let's try something:

x^2-6x =216
take half of the coefficient of x, the -6. that's -3 , square that and add to both sides of the equation
x^2-6x+9=216+9
(x-3)^2 = 225 take sqrt of both sides
x-3=+-sqrt(225)
x=3+-sqrt(225)
x=3 +-15
x=18 and x=-12
We found the numbers ( actually solved the equation by what is called "completing the square". Do the same thing on a general equation "ax^2 +bx+c=0 But first divide both sides by a x^2+(b/a)x +c/a=0 and you will derive the quadatic formula. Show your teacher for a cinch A.

Answer 4

Yes, by definition, the quad eqn has to equal zero.
Your problem:
x^2 - 6x - 216 = 0
Try to begin by small numbers, like 2 or 4; let us try 4:
216 = 4 x 54
54 = 9 x 6
so 216 = 4 x 9 x 6 = 2x2 x 3x3 x 3x2
So try
(x+12)(x-18) = x^2 + 12x - 18x + 216 that got it
You see what I did? I took the 2x2x3 = 12 and then the 3x3x3x2 = 18; then I noticed that -18 + 12 = -6.
It takes practice and lots of factors, factors, factors.

Answer 5

One way is to break the unit (non-x) number down into its prime factors first. This can make it easier to spot which pairs of numbers are available to you.

For example, 216 = 2^3 x 3^3
Splitting these into 2 x 3^2 and 2^2 x 3 gives 18 and 12, as mentioned in the previous answers.

Answer 6

In order to use the quadratic formula the one side has to equals 0
x^2-6x-216=0
(x-18)(x+2)=0
x=18 x=-2

Answer 7

u just have to be patient with factoring sometimes
x^2-6x-216=0
(x+12)(x-18)=0
x+12=0 x-18=0
x=-12 x=18

Answer 8

a quadratic needs to be st to zero

x^2 -6x -216 = 0

(x + 12)(x - 18)

x = -12 or x = 18

Answer 9

notice 216 = 18*12, and 18-12 = 6 You can factor it now.

Answer 10

x² -6x = 216
x² -6x -216 = 0
(x-18)(x+12)=0
x={18,-12}

There's your factoring!