Answer: x = 5
x = -4
x = 0
Step-by-step explanation:The first term is, x2 its coefficient is 1 .
The middle term is, -x its coefficient is -1 .
The last term, "the constant", is -20
Step-1 : Multiply the coefficient of the first term by the constant 1 • -20 = -20
Step-2 : Find two factors of -20 whose sum equals the coefficient of the middle term, which is -1 .
-20 + 1 = -19
-10 + 2 = -8
-5 + 4 = -1 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -5 and 4
x2 - 5x + 4x - 20
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x-5)
Add up the last 2 terms, pulling out common factors :
4 • (x-5)
Step-5 : Add up the four terms of step 4 :
(x+4) • (x-5)
Which is the desired factorization A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well. Solving x2-x-20 = 0 by Completing The Square .
Add 20 to both side of the equation :
x2-x = 20
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 :
20 + 1/4 or, (20/1)+(1/4)
The common denominator of the two fractions is 4 Adding (80/4)+(1/4) gives 81/4
So adding to both sides we finally get :
x2-x+(1/4) = 81/4
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) = 81/4 and
x2-x+(1/4) = (x-(1/2))2
then, according to the law of transitivity,
(x-(1/2))2 = 81/4
We'll refer to this Equation as Eq. #4.2.1
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. #4.2.1 we get:
x-(1/2) = √ 81/4
Add 1/2 to both sides to obtain:
x = 1/2 + √ 81/4
Since a square root has two values, one positive and the other negative
x2 - x - 20 = 0
has two solutions:
x = 1/2 + √ 81/4
or
x = 1/2 - √ 81/4
Note that √ 81/4 can be written as
√ 81 / √ 4 which is 9 / 2