Solution - Factoring binomials using the difference of squares
Step by Step Solution
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
16*d^2-(4)=0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
24d2 - 4 = 0
Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
16d2 - 4 = 4 • (4d2 - 1)
Trying to factor as a Difference of Squares :
3.2 Factoring: 4d2 - 1
Theory : A difference of two perfect squares, A2 - B2 can be factored into (A+B) • (A-B)
Proof : (A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : AB = BA is the commutative property of multiplication.
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check : 4 is the square of 2
Check : 1 is the square of 1
Check : d2 is the square of d1
Factorization is : (2d + 1) • (2d - 1)
Equation at the end of step 3 :
4 • (2d + 1) • (2d - 1) = 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 : 4 = 0
This equation has no solution.
A a non-zero constant never equals zero.
Solving a Single Variable Equation :
4.3 Solve : 2d+1 = 0
Subtract 1 from both sides of the equation :
2d = -1
Divide both sides of the equation by 2:
d = -1/2 = -0.500
Solving a Single Variable Equation :
4.4 Solve : 2d-1 = 0
Add 1 to both sides of the equation :
2d = 1
Divide both sides of the equation by 2:
d = 1/2 = 0.500
Two solutions were found :
- d = 1/2 = 0.500
- d = -1/2 = -0.500
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