STEP
1
:
Equation at the end of step 1
(3x2 + 36x) + 81 = 0
STEP
2
:
STEP
3
:
Pulling out like terms
3.1 Pull out like factors :
3x2 + 36x + 81 = 3 • (x2 + 12x + 27)
Trying to factor by splitting the middle term
3.2 Factoring x2 + 12x + 27
The first term is, x2 its coefficient is 1 .
The middle term is, +12x its coefficient is 12 .
The last term, "the constant", is +27
Step-1 : Multiply the coefficient of the first term by the constant 1 • 27 = 27
Step-2 : Find two factors of 27 whose sum equals the coefficient of the middle term, which is 12 .
-27 + -1 = -28
-9 + -3 = -12
-3 + -9 = -12
-1 + -27 = -28
1 + 27 = 28
3 + 9 = 12 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 3 and 9
x2 + 3x + 9x + 27
Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+3)
Add up the last 2 terms, pulling out common factors :
9 • (x+3)
Step-5 : Add up the four terms of step 4 :
(x+9) • (x+3)
Which is the desired factorization
Equation at the end of step
3
:
3 • (x + 9) • (x + 3) = 0
STEP
4
:
Theory - Roots of a product
4.1 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.
Equations which are never true:
4.2 Solve : 3 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solving a Single Variable Equation:
4.3 Solve : x+9 = 0
Subtract 9 from both sides of the equation :
x = -9
Solving a Single Variable Equation:
4.4 Solve : x+3 = 0
Subtract 3 from both sides of the equation :
x = -3
Supplement : Solving Quadratic Equation Directly
Solving x2+12x+27 = 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
Step-by-step explanation:
