What is. the partial fraction for 7x-1 divided by xsquare -5x+6?
Answers
Two solutions were found :
x = 1/4 = 0.250
x = 1/3 = 0.333
Step by step solution :
Step 1 :
Equation at the end of step 1 :
((0 - (22•3x2)) + 7x) - 1 = 0
Step 2 :
Step 3 :
Pulling out like terms :
3.1 Pull out like factors :
-12x2 + 7x - 1 = -1 • (12x2 - 7x + 1)
Trying to factor by splitting the middle term
3.2 Factoring 12x2 - 7x + 1
The first term is, 12x2 its coefficient is 12 .
The middle term is, -7x its coefficient is -7 .
The last term, "the constant", is +1
Step-1 : Multiply the coefficient of the first term by the constant 12 • 1 = 12
Step-2 : Find two factors of 12 whose sum equals the coefficient of the middle term, which is -7 .
-12 + -1 = -13
-6 + -2 = -8
-4 + -3 = -7 That's it
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and -3
12x2 - 4x - 3x - 1
Step-4 : Add up the first 2 terms, pulling out like factors :
4x • (3x-1)
Add up the last 2 terms, pulling out common factors :
1 • (3x-1)
Step-5 : Add up the four terms of step 4 :
(4x-1) • (3x-1)
Which is the desired factorization
Equation at the end of step 3 :
(1 - 3x) • (4x - 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.
Solving a Single Variable Equation :
4.2 Solve : -3x+1 = 0
Subtract 1 from both sides of the equation :
-3x = -1
Multiply both sides of the equation by (-1) : 3x = 1
Divide both sides of the equation by 3:
x = 1/3 = 0.333
Solving a Single Variable Equation :
4.3 Solve : 4x-1 = 0
Add 1 to both sides of the equation :
4x = 1
Divide both sides of the equation by 4:
x = 1/4 = 0.250
Supplement : Solving Quadratic Equation Directly
Solving 12x2-7x+1 = 0 directly
y = 12.0 * 0.29 * 0.29 - 7.0 * 0.29 + 1.0
or y = -0.021
Root plot for : y = 12x2-7x+1
Axis of Symmetry (dashed) {x}={ 0.29}
Vertex at {x,y} = { 0.29,-0.02}
x -Intercepts (Roots) :
Root 1 at {x,y} = { 0.25, 0.00}
Root 2 at {x,y} = { 0.33, 0.00}
Solve Quadratic Equation by Completing The Square
5.2 Solving 12x2-7x+1 = 0 by Completing The Square .
Divide both sides of the equation by 12 to have 1 as the coefficient of the first term :
x2-(7/12)x+(1/12) = 0
Subtract 1/12 from both side of the equation :
x2-(7/12)x = -1/12
Now the clever bit: Take the coefficient of x , which is 7/12 , divide by two, giving 7/24 , and finally square it giving 49/576
Add 49/576 to both sides of the equation :
On the right hand side we have :
-1/12 + 49/576 The common denominator of the two fractions is 576 Adding (-48/576)+(49/576) gives 1/576
So adding to both sides we finally get :
x2-(7/12)x+(49/576) = 1/576
Adding 49/576 has completed the left hand side into a perfect square :
x2-(7/12)x+(49/576) =
(x-(7/24)) • (x-(7/24)) =
(x-(7/24))2
Things which are equal to the same thing are also equal to one another. Since
x2-(7/12)x+(49/576) = 1/576 and
x2-(7/12)x+(49/576) = (x-(7/24))2
then, according to the law of transitivity,
(x-(7/24))2 = 1/576
We'll refer to this Equation as Eq. #5.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(7/24))2 is
(x-(7/24))2/2 =
(x-(7/24))1 =
x-(7/24)
Now, applying the Square Root Principle to Eq. #5.2.1 we get:
x-(7/24) = √ 1/576
Add 7/24 to both sides to obtain:
x = 7/24 + √ 1/576
Since a square root has two values, one positive and the other negative
x2 - (7/12)x + (1/12) = 0
has two solutions:
x = 7/24 + √ 1/576
or
x = 7/24 - √ 1/576
Note that √ 1/576 can be written as
√ 1 / √ 576 which is 1 / 24
Solve Quadratic Equation using the Quadratic Formula
5.3 Solving 12x2-7x+1 = 0 by the Quadratic Formula .
According to the Quadratic Formula, x , the solution for Ax2+Bx+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
x = ————————
2A
In our case, A = 12
B = -7
C = 1
Accordingly, B2 - 4AC =
49 - 48 =
1
Applying the quadratic formula :
7 ± √ 1
x = ————
24
So now we are looking at:
x = ( 7 ± 1) / 24
Two real solutions:
x =(7+√1)/24= 0.333
or:
x =(7-√1)/24= 0.250
Two solutions were found :
x = 1/4 = 0.250
x = 1/3 = 0.333
Processing ends successfully
Step-by-step explanation:
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