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This solution deals with simplification or other simple results.
- Simplification or other simple results
((0 - 5x2) - 35x) - 50
3.1 Pull out like factors :
-5x2 - 35x - 50 = -5 • (x2 + 7x + 10)
Trying to factor by splitting the middle term
3.2 Factoring x2 + 7x + 10
The first term is, x2 its coefficient is 1 .
The middle term is, +7x its coefficient is 7 .
The last term, "the constant", is +10
Step-1 : Multiply the coefficient of the first term by the constant 1 • 10 = 10
Step-2 : Find two factors of 10 whose sum equals the coefficient of the middle term, which is 7 .
-10 | + | -1 | = | -11 | ||
-5 | + | -2 | = | -7 | ||
-2 | + | -5 | = | -7 | ||
-1 | + | -10 | = | -11 | ||
1 | + | 10 | = | 11 | ||
2 | + | 5 | = | 7 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 2 and 5
x2 + 2x + 5x + 10Step-4 : Add up the first 2 terms, pulling out like factors :
x • (x+2)
Add up the last 2 terms, pulling out common factors :5 • (x+2)
Step-5 : Add up the four terms of step 4 :(x+5) • (x+2)
Which is the desired factorization
Final result :
-5 • (x + 5) • (x + 2)
Changes made to your input should not affect the solution:
(1): "x5" was replaced by "x^5".
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(5 • (x2)) - (5•7x50) = 0Equation at the end of step 2 :
5x2 - (5•7x50) = 0Step 3 :
Step 4 :
Pulling out like terms :
4.1 Pull out like factors :
5x2 - 35x50 = -5x2 • (7x48 - 1)
Trying to factor as a Difference of Squares :
4.2 Factoring: 7x48 - 1
Theory : A difference of two perfect squares, A2 - B2 can be factored into Proof :
(A+B) • (A-B) =
A2 - AB + BA - B2 =
A2 - AB + AB - B2 =
A2 - B2
Note : - AB + AB equals zero and is therefore eliminated from the expression.
Check :Ruling : Binomial can not be factored as the
difference of two perfect squaresTrying to factor as a Difference of Cubes:
4.3 Factoring: 7x48 - 1
Theory : A difference of two perfect cubes, a3 - b3 can be factored into
(a-b) • (a2 +ab +b2)
Proof : (a-b)•(a2+ab+b2) =
a3+a2b+ab2-ba2-b2a-b3 =
a3+(a2b-ba2)+(ab2-b2a)-b3 =
a3+0+0-b3 =
a3-b3
Check : 7 is not a cube !!
Ruling : Binomial can not be factored as the difference of two perfect cubes
Equation at the end of step 4 :
-5x2 • (7x48 - 1) = 0Step 5 :
Theory - Roots of a product :
5.1 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 separatelyIn other words, we are going to solve as many equations as there are terms in the productAny solution of term = 0 solves product = 0 as well.
Solving a Single Variable Equation :
5.2 Solve : -5x2 = 0Multiply both sides of the equation by (-1) : 5x2 = 0 Divide both sides of the equation by 5:
x2 = 0
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ 0 Any root of zero is zero. This equation has one solution which is x = 0
Solving a Single Variable Equation :
5.3 Solve : 7x48-1 = 0Add 1 to both sides of the equation :
7x48 = 1 Divide both sides of the equation by 7:
x48 = 1/7 = 0.143
x = 48th root of (1/7)The equation has two real solutions
These solutions are x = 48th root of ( 0.143) = ± 0.96027
Three solutions were found :
- x = 48th root of ( 0.143) = ± 0.96027
- x = 0
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