x^2-14x+24=0

Hello Emma! Your equation is quadratic, that is of the form ax^2 + bx + c = 0

where a = 1, b = -14,  and c = 24.      so we have: x^2 - 14x + 24 = 0

since a = 1 we factor c ( 24)  and we add the factors to see if they add up to b (-14).

notice that c is positive =>this means that the two factors were looking for have the same sign.

also since b is negative we know that the larger and hence both of the factors in question are negative.

Now we seek two negative numbers whose product is 24 and whose sum is -14.

fgactoring 24 we get: -1*-24 whose sum is -25 no good. next we try -2*-12 whose sum is -14 <= BINGO!

Now we write:  x^2 - 14x + 24 =0

                      (x - 2) (x - 12) = 0

so set each factor to zero:  x-2 = 0 ; x - 12 =0

Finally solve the two resulting linear equations to get: x = 2 ,and x = 12.

Hi Emma, 

I am a math PhD from UW and while both of these earlier answers are good, they show that x^2 - 14x + 24=0 was an unfair question on 8/30, so early in Algebra I!  

Maybe it was a pre-test to see how much factoring you got in Pre-algebra.... Not enough, right?  :+)

Let's see; we need  x^2 - 14x + 24 to factor into (x - d) *(x -e) so that 

d*e is 24 and d + e is 14, because x^2  - (d+e)*x + d*e must equal the given left side

and the factors of 24 that work are d=12 and e=2

[we know both linear factors have  minus signs because of the -14 and the +24]

so (x - 12)*(x - 2) = 0, which makes both linear factors  = zero: x -12 = 0  &  x-2=0,

so x = 12 and x = 2 are the quadratic's solutions.

Altho this question is long past, if you have similar ones in future, shoot 'em to me, nearby in Poulsbo, WA.

/Doc Fred

Quadratic Equestion: ax² + bx + c = 0

our equestion : x² - 14x + 24 =0

therefore, a=1, b= -14, c= 24

Factor c to get, product= 24 and sum= -14  (c is +ve:  two factors were looking for have the same sign.)

factor: (-2) and (-12)  which has sum= -14 and product= 24

x² - 14x + 24 =0

x²-12x-2x+24=0

x(x - 12) - 2(x -  12) = 0

(x- 12)* (x- 2) = 0

x- 12=0 and  x-2= 0

solvinf the linear equations we get,

x=12 and x=2