Simple Equations

Assuming $\ x \ne 7$ $\frac{x^2-x-42}{x-7} = \frac{(x-7)(x+6)}{x-7} = x+6 = 0 \therefore\ x = -6$

Simple way to answer this type quadratic equation. If any likes it then reply me.
Consider the numerator as
Now, focus on the term -42, which can be factorized into -7 and 6 or 7 and -6. But, according to the term -x, the factors must be -7 and 6.
Hence, we will get
(x-7)(x+6) = 0. Putting this in the equation, we get.....

Everybody has a correct answer, however nobody expresses a general algebraic law on how to come up with the factors -7 and 6 of the number -42 (which could also be factorized into -2 and 21, or -14 and 3 ).

Here's my attempt to express a more general algebraic law to get the factors -7 and 6 (Hope you like it):

Any quadratic expression of the form

a(x^2) + bx + c = 0 , ............. (Eq. 1)

can be factorized into the form :

(x + d) * (x + e),............... (Eq. 2)

in which:

d + e = b ................(Eq. 3) ......("b" from Eq. 1)

and

d * e = c ..................(Eq. 4).......("c" from Eq. 1)

d and e are two "unknowns" in a system of two equations (Eqs. 3 and 4). therefore they can be solved by substitution :

Solving Eq. 4 for e :

e = b - d

Substituting e in Eq. 3:

d * (b-d) = c

or

-(d^2)+ (bd)- c = 0 .............. (Eq. 5)

Solving Eq. 5 with the quadratic formula will return two possible values for d, however this ambiguity is easily solved by substituting the two possible values of d in order to satisfy Eq 3. Try it with the example here, d = -7 and e = 6 .

In order for the quotient to be zero, the numerator needs to be equal to zero. But x cannot be equal to zero, because denominators cannot be zero.

So x cannot be equal to 7 since that will make the denominator zero.

$x^{2}$ - x - 42 = 0

(x-7) (x+6) = 0

x = 7 or x = -6

Since x cannot be equal to 7, x = -6.

Therefore, the solution to the equation ( $x^{2}$ - x - 42) / (x - 7) = 0 is x = -6.

Just simply solve the quadratic equation x^2 - x - 42 , and you'll get -6 or 7. but 7 can't be the answer because if so, the answer will be undefined.

x^2 - x - 42 = 0

( x+6 ) ( x -7) = 0

x = -6 or 7

(x-7)(x+6) _ _ =x+6 (x-7)

x+6=0 x= -6 N.B. Assuming x≠7

quardetric equations .....

(x^2 - x - 42)/(x - 7) = [(x - 7)(x + 6)]/(x-7) = (x +6) = 0, x = -6, -6 + 6=0

Just factorize it.

(x+6)(x-7)/(x-7)=0 => x+6=0 =>x=-6

(x^2 - x - 42)/(x-7) = 0

from here it is already clear that (x-7) <> 0, otherwise the equation will not be equal to zero.

Factoring the equation

(x+6)(x-7)/(x-7) = 0

Since (x - 7) <> 0, cancellation law applies to the equation. Therefore

(x+6) = 0

x = -6