Question

The factorisation x2 + x +  14 is​

Answer 1

Answer:

x²-5x−14=0

x

2

−7x+2x−14=0

x(x−7)+2(x−7)=0

(x−7)(x+2)=0

(x−7)(x+2)=0

x=7,−2

Answer 2

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Add 14 to both side of the equation :

x2-x = 14

Now the clever bit: Take the coefficient of x , which is 1 , divide by two, giving 1/2 , and finally square it giving 1/4

Add 1/4 to both sides of the equation :

On the right hand side we have :

14 + 1/4 or, (14/1)+(1/4)

The common denominator of the two fractions is 4 Adding (56/4)+(1/4) gives 57/4

So adding to both sides we finally get :

x2-x+(1/4) = 57/4

Adding 1/4 has completed the left hand side into a perfect square :

x2-x+(1/4) =

(x-(1/2)) • (x-(1/2)) =

(x-(1/2))2

Things which are equal to the same thing are also equal to one another. Since

x2-x+(1/4) = 57/4 and

x2-x+(1/4) = (x-(1/2))2

then, according to the law of transitivity,

(x-(1/2))2 = 57/4

We'll refer to this Equation as Eq. #2.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of

(x-(1/2))2 is

(x-(1/2))2/2 =

(x-(1/2))1 =

x-(1/2)

Now, applying the Square Root Principle to Eq. #2.2.1 we get:

x-(1/2) = √ 57/4

Add 1/2 to both sides to obtain:

x = 1/2 + √ 57/4

Since a square root has two values, one positive and the other negative

x2 - x - 14 = 0

has two solutions:

x = 1/2 + √ 57/4

or

x = 1/2 - √ 57/4

Note that √ 57/4 can be written as

√ 57 / √ 4 which is √ 57 / 2