Simplify: -(24)/(y^(2)+3y-28)=-(2y)/(y^(2)+7) Tiger Algebra Solver (2024)

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

-(24)/(y^(2)+3*y-28)-(-(2*y)/(y^(2)+7))=0

1.1 Find roots (zeroes) of : F(y) = y² + 7
Polynomial Roots Calculator is a set of methods aimed at finding values ofyfor which F(y)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers y which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 1 and the Trailing Constant is 7. The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,7 Let us test ....

2.1Factoring y² + 3y - 28

The first term is, y² its coefficient is 1.
The middle term is, +3y its coefficient is 3.
The last term, "the constant", is -28

Step-1 : Multiply the coefficient of the first term by the constant 1•-28=-28

Step-2 : Find two factors of -28 whose sum equals the coefficient of the middle term, which is 3.

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -4 and 7
y² - 4y+7y - 28

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:
7•(y-4)
Step-5:Add up the four terms of step4:
(y+7)•(y-4)
Which is the desired factorization

3.1 Find the Least Common Multiple

The left denominator is : (y+7)•(y-4)

The right denominator is : y²+7

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=y²+7

Right_M=L.C.M/R_Deno=(y+7)•(y-4)

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:

2y³ - 18y² - 56y - 168=

2•(y³ - 9y² - 28y - 84)

4.3 Factoring: y³ - 9y² - 28y - 84

Thoughtfully split the expression at hand into groups, each group having two terms:

Group 1: y³ - 9y²
Group 2: -28y - 84

Pull out from each group separately :

Group 1: (y - 9) • (y²)
Group 2: (y + 3) • (-28)

Bad news !! Factoring by pulling out fails : The groups have no common factor and can not be added up to form a multiplication.

4.4 Find roots (zeroes) of : F(y) = y³ - 9y² - 28y - 84

See theory in step 1.1
In this case, the Leading Coefficient is 1 and the Trailing Constant is -84. The factor(s) are:

of the Leading Coefficient : 1
of the Trailing Constant : 1 ,2 ,3 ,4 ,6 ,7 ,12 ,14 ,21 ,28 , etc Let us test ....

Note - For tidiness, printing of 15 checks which found no root was suppressed

Polynomial Roots Calculator found no rational roots

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+7)•(y-4)•(y²+7) cancels out the denominator, while, on the right hand side, zero times anything is still zero.

The equation now takes the shape:
2 • (y³-9y²-28y-84)=0

5.2Solve:2=0

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

5.3Solvey³-9y²-28y-84 = 0

Future releases of Tiger-Algebra will solve equations of the third degree directly.

Meanwhile we will use the Bisection method to approximate one real solution.

We now use the Bisection Method to approximate one of the solutions. The Bisection Method is an iterative procedure to approximate a root (Root is another name for a solution of an equation).

The function is F(y) = y³ - 9y² - 28y - 84

Aty= 11.00 F(y) is equal to -150.00
Aty= 12.00 F(y) is equal to 12.00

Intuitively we feel, and justly so, that since F(y) is negative on one side of the interval, and positive on the other side then, somewhere inside this interval, F(y) is zero

Procedure :
(1) Find a point "Left" where F(Left) < 0

(2) Find a point 'Right' where F(Right) > 0

(3) Compute 'Middle' the middle point of the interval [Left,Right]

(4) Calculate Value = F(Middle)

(5) If Value is close enough to zero goto Step (7)

Else :
If Value < 0 then : Left <- Middle
If Value > 0 then : Right <- Middle

(6) Loop back to Step (3)

(7) Done!! The approximation found is Middle

Follow Middle movements to understand how it works :

Next Middle will get us close enough to zero:

F(11.935575552) is 0.000000191

The desired approximation of the solution is:

y ≓ 11.935575552

Note, ≓ is the approximation symbol

Root plot for : t = y²+3y-28
Axis of Symmetry (dashed) {y}={-1.50}
Vertex at {y,t} = {-1.50,-30.25}
y-Intercepts (Roots) :
Root 1 at {y,t} = {-7.00, 0.00}
Root 2 at {y,t} = { 4.00, 0.00}

6.2Solvingy²+3y-28 = 0 by Completing The Square.Add 28 to both side of the equation :
y²+3y = 28

Now the clever bit: Take the coefficient of y, which is 3, divide by two, giving 3/2, and finally square it giving 9/4

Add 9/4 to both sides of the equation :
On the right hand side we have:
28+9/4or, (28/1)+(9/4)
The common denominator of the two fractions is 4Adding (112/4)+(9/4) gives 121/4
So adding to both sides we finally get:
y²+3y+(9/4) = 121/4

Adding 9/4 has completed the left hand side into a perfect square :
y²+3y+(9/4)=
(y+(3/2))•(y+(3/2))=
(y+(3/2))²
Things which are equal to the same thing are also equal to one another. Since
y²+3y+(9/4) = 121/4 and
y²+3y+(9/4) = (y+(3/2))²
then, according to the law of transitivity,
(y+(3/2))² = 121/4

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+(3/2))² is
(y+(3/2))^2/2=
(y+(3/2))¹=
y+(3/2)

Now, applying the Square Root Principle to Eq.#6.2.1 we get:
y+(3/2)= √ 121/4

Subtract 3/2 from both sides to obtain:
y = -3/2 + √ 121/4

Since a square root has two values, one positive and the other negative
y² + 3y - 28 = 0
has two solutions:
y = -3/2 + √ 121/4
or
y = -3/2 - √ 121/4

Note that √ 121/4 can be written as
√121 / √4which is 11 / 2

6.3Solvingy²+3y-28 = 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= 3
C=-28 Accordingly,B²-4AC=
9 - (-112) =
121Applying the quadratic formula :

-3 ± √ 121
y=——————
2Can √ 121 be simplified ?

Yes!The prime factorization of 121is
11•11
To be able to remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. second root).

√ 121 =√11•11 =
±11 •√ 1 =
±11

So now we are looking at:
y=(-3±11)/2

Two real solutions:

y =(-3+√121)/2=(-3+11)/2= 4.000

or:

y =(-3-√121)/2=(-3-11)/2= -7.000