Simplify: ((4y)/(y+6))=-(((16)/(((y)^(2))+2y-24))) Tiger Algebra Solver (2024)

Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :

((4*y)/(y+6))-(-(((16)/(((y)^(2))+2*y-24))))=0

1.1Factoring y² + 2y - 24

The first term is, y² its coefficient is 1.
The middle term is, +2y its coefficient is 2.
The last term, "the constant", is -24

Step-1 : Multiply the coefficient of the first term by the constant 1•-24=-24

Step-2 : Find two factors of -24 whose sum equals the coefficient of the middle term, which is 2.

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -4 and 6
y² - 4y+6y - 24

Step-4 : Add up the first 2 terms, pulling out like factors:
y•(y-4)
Add up the last 2 terms, pulling out common factors:
6•(y-4)
Step-5:Add up the four terms of step4:
(y+6)•(y-4)
Which is the desired factorization

3.1 Find the Least Common Multiple

The left denominator is : y+6

The right denominator is : (y+6)•(y-4)

3.2 Calculate multipliers for the two fractions

Denote the Least Common Multiple by L.C.M
Denote the Left Multiplier by Left_M
Denote the Right Multiplier by Right_M
Denote the Left Deniminator by L_Deno
Denote the Right Multiplier by R_Deno

Left_M=L.C.M/L_Deno=y-4

Right_M=L.C.M/R_Deno=1

3.3 Rewrite the two fractions into equivalent fractions

Two fractions are called equivalent if they have the same numeric value.

For example : 1/2 and 2/4 are equivalent, y/(y+1)² and (y²+y)/(y+1)³ are equivalent as well.

To calculate equivalent fraction , multiply the Numerator of each fraction, by its respective Multiplier.

3.4 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:

4.1 Pull out like factors:

4y² - 16y + 16=4•(y² - 4y + 4)

4.2Factoring y² - 4y + 4

The first term is, y² its coefficient is 1.
The middle term is, -4y its coefficient is -4.
The last term, "the constant", is +4

Step-1 : Multiply the coefficient of the first term by the constant 1•4=4

Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is -4.

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -2 and -2
y² - 2y-2y - 4

Step-4 : Add up the first 2 terms, pulling out like factors:
y•(y-2)
Add up the last 2 terms, pulling out common factors:
2•(y-2)
Step-5:Add up the four terms of step4:
(y-2)•(y-2)
Which is the desired factorization

4.3 Multiply (y-2) by (y-2)

The rule says : To multiply exponential expressions which have the same base, add up their exponents.

In our case, the common base is (y-2) and the exponents are:
1,as(y-2) is the same number as (y-2)¹
and1,as(y-2) is the same number as (y-2)¹
The product is therefore, (y-2)⁽¹⁺¹⁾ = (y-2)²

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:

Now, on the left hand side, the (y+6)•(y-4) cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape:
4 • (y-2)² =0

5.2Solve:4=0

This equation has no solution.
A a non-zero constant never equals zero.

5.3Solve:(y-2)² = 0(y-2)² represents, in effect, a product of 2 terms which is equal to zero

For the product to be zero, at least one of these terms must be zero. Since all these terms are equal to each other, it actually means: y-2=0

Add 2 to both sides of the equation:
y = 2

Root plot for : t = y²-4y+4
Vertex at {y,t} = { 2.00, 0.00}
y-Intercept (Root) :
One Root at {y,t}={ 2.00, 0.00}
Note that the root coincides with
the Vertex and the Axis of Symmetry
coinsides with the line y = 0

6.2Solvingy²-4y+4 = 0 by Completing The Square.Subtract 4 from both side of the equation :
y²-4y = -4

Now the clever bit: Take the coefficient of y, which is 4, divide by two, giving 2, and finally square it giving 4

Add 4 to both sides of the equation :
On the right hand side we have:
-4+4or, (-4/1)+(4/1)
The common denominator of the two fractions is 1Adding (-4/1)+(4/1) gives 0/1
So adding to both sides we finally get:
y²-4y+4 = 0

Adding 4 has completed the left hand side into a perfect square :
y²-4y+4=
(y-2)•(y-2)=
(y-2)²
Things which are equal to the same thing are also equal to one another. Since
y²-4y+4 = 0 and
y²-4y+4 = (y-2)²
then, according to the law of transitivity,
(y-2)² = 0

We'll refer to this Equation as Eq. #6.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
(y-2)² is
(y-2)^2/2=
(y-2)¹=
y-2

Now, applying the Square Root Principle to Eq.#6.2.1 we get:
y-2= √ 0

Add 2 to both sides to obtain:
y = 2 + √ 0
The square root of zero is zero

This quadratic equation has one solution only. That's because adding zero is the same as subtracting zero.

The solution is:
y = 2

6.3Solvingy²-4y+4 = 0 by the Quadratic Formula.According to the Quadratic Formula,y, the solution forAy²+By+C= 0 , where A, B and C are numbers, often called coefficients, is given by :

-B± √B²-4AC
y = ————————
2A In our case,A= 1
B= -4
C= 4 Accordingly,B²-4AC=
16 - 16 =
0Applying the quadratic formula :

4 ± √ 0
y=————
2The square root of zero is zero

This quadratic equation has one solution only. That's because adding zero is the same as subtracting zero.

The solution is:
y = 4 / 2 = 2