Answer:
Explanation:
Step by step solution :
STEP
1
:
5
Simplify —
9
Equation at the end of step
1
:
5
((9 • (x2)) - 6x) + — = 0
9
STEP
2
:
Equation at the end of step
2
:
5
(32x2 - 6x) + — = 0
9
STEP
3
:
Rewriting the whole as an Equivalent Fraction
3.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 9 as the denominator :
9x2 - 6x (9x2 - 6x) • 9
9x2 - 6x = ———————— = ——————————————
1 9
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
STEP
4
:
Pulling out like terms
4.1 Pull out like factors :
9x2 - 6x = 3x • (3x - 2)
Adding fractions that have a common denominator :
4.2 Adding up the two equivalent fractions
Add the two equivalent fractions which now have a common denominator
Combine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:
3x • (3x-2) • 9 + 5 81x2 - 54x + 5
——————————————————— = ——————————————
9 9
Trying to factor by splitting the middle term
4.3 Factoring 81x2 - 54x + 5
The first term is, 81x2 its coefficient is 81 .
The middle term is, -54x its coefficient is -54 .
The last term, "the constant", is +5
Step-1 : Multiply the coefficient of the first term by the constant 81 • 5 = 405
Step-2 : Find two factors of 405 whose sum equals the coefficient of the middle term, which is -54 .
-405 + -1 = -406
-135 + -3 = -138
-81 + -5 = -86
-45 + -9 = -54 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -45 and -9
81x2 - 45x - 9x - 5
Step-4 : Add up the first 2 terms, pulling out like factors :
9x • (9x-5)
Add up the last 2 terms, pulling out common factors :
1 • (9x-5)
Step-5 : Add up the four terms of step 4 :
(9x-1) • (9x-5)
Which is the desired factorization
Equation at the end of step
4
:
(9x - 5) • (9x - 1)
——————————————————— = 0
9
STEP
5
:
When a fraction equals zero :
5.1 When a fraction equals zero ...
Where a fraction equals zero, its numerator, the part which is above the fraction line, must equal zero.
Now,to get rid of the denominator, Tiger multiplys both sides of the equation by the denominator.
Here's how:
(9x-5)•(9x-1)
————————————— • 9 = 0 • 9
9
Now, on the left hand side, the 9 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
(9x-5) • (9x-1) = 0
Theory - Roots of a product :
5.2 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 a Single Variable Equation:
5.3 Solve : 9x-5 = 0
Add 5 to both sides of the equation :
9x = 5
Divide both sides of the equation by 9:
x = 5/9 = 0.556
Solving a Single Variable Equation:
5.4 Solve : 9x-1 = 0
Add 1 to both sides of the equation :
9x = 1
Divide both sides of the equation by 9:
x = 1/9 = 0.111
Supplement : Solving Quadratic Equation Directly
Solving 81x2-54x+5 = 0 directly
Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula