Solution - Adding, subtracting and finding the least common multiple
Other Ways to Solve
Adding, subtracting and finding the least common multipleStep by Step Solution
Step by step solution :
Step 1 :
9
Simplify ——
16
Equation at the end of step 1 :
3 9 ((x4)+(—•(x2)))+—— = 0 2 16Step 2 :
3 Simplify — 2
Equation at the end of step 2 :
3 9
((x4) + (— • x2)) + —— = 0
2 16
Step 3 :
Equation at the end of step 3 :
3x2 9 ((x4) + ———) + —— = 0 2 16Step 4 :
Rewriting the whole as an Equivalent Fraction :
4.1 Adding a fraction to a whole
Rewrite the whole as a fraction using 2 as the denominator :
x4 x4 • 2
x4 = —— = ——————
1 2
Equivalent fraction : The fraction thus generated looks different but has the same value as the whole
Common denominator : The equivalent fraction and the other fraction involved in the calculation share the same denominator
Adding fractions that have a common denominator :
4.2 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:
x4 • 2 + 3x2 2x4 + 3x2
———————————— = —————————
2 2
Equation at the end of step 4 :
(2x4 + 3x2) 9
——————————— + —— = 0
2 16
Step 5 :
Step 6 :
Pulling out like terms :
6.1 Pull out like factors :
2x4 + 3x2 = x2 • (2x2 + 3)
Polynomial Roots Calculator :
6.2 Find roots (zeroes) of : F(x) = 2x2 + 3
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/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 2 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,2
of the Trailing Constant : 1 ,3
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 5.00 | ||||||
| -1 | 2 | -0.50 | 3.50 | ||||||
| -3 | 1 | -3.00 | 21.00 | ||||||
| -3 | 2 | -1.50 | 7.50 | ||||||
| 1 | 1 | 1.00 | 5.00 | ||||||
| 1 | 2 | 0.50 | 3.50 | ||||||
| 3 | 1 | 3.00 | 21.00 | ||||||
| 3 | 2 | 1.50 | 7.50 |
Polynomial Roots Calculator found no rational roots
Calculating the Least Common Multiple :
6.3 Find the Least Common Multiple
The left denominator is : 2
The right denominator is : 16
| Prime Factor | Left Denominator | Right Denominator | L.C.M = Max {Left,Right} |
|---|---|---|---|
| 2 | 1 | 4 | 4 |
| Product of all Prime Factors | 2 | 16 | 16 |
Least Common Multiple:
16
Calculating Multipliers :
6.4 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 = 8
Right_M = L.C.M / R_Deno = 1
Making Equivalent Fractions :
6.5 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.
L. Mult. • L. Num. x2 • (2x2+3) • 8 —————————————————— = ———————————————— L.C.M 16 R. Mult. • R. Num. 9 —————————————————— = —— L.C.M 16
Adding fractions that have a common denominator :
6.6 Adding up the two equivalent fractions
x2 • (2x2+3) • 8 + 9 16x4 + 24x2 + 9
———————————————————— = ———————————————
16 16
Trying to factor by splitting the middle term
6.7 Factoring 16x4 + 24x2 + 9
The first term is, 16x4 its coefficient is 16 .
The middle term is, +24x2 its coefficient is 24 .
The last term, "the constant", is +9
Step-1 : Multiply the coefficient of the first term by the constant 16 • 9 = 144
Step-2 : Find two factors of 144 whose sum equals the coefficient of the middle term, which is 24 .
| -144 | + | -1 | = | -145 | ||
| -72 | + | -2 | = | -74 | ||
| -48 | + | -3 | = | -51 | ||
| -36 | + | -4 | = | -40 | ||
| -24 | + | -6 | = | -30 | ||
| -18 | + | -8 | = | -26 | ||
| -16 | + | -9 | = | -25 | ||
| -12 | + | -12 | = | -24 | ||
| -9 | + | -16 | = | -25 | ||
| -8 | + | -18 | = | -26 | ||
| -6 | + | -24 | = | -30 | ||
| -4 | + | -36 | = | -40 | ||
| -3 | + | -48 | = | -51 | ||
| -2 | + | -72 | = | -74 | ||
| -1 | + | -144 | = | -145 | ||
| 1 | + | 144 | = | 145 | ||
| 2 | + | 72 | = | 74 | ||
| 3 | + | 48 | = | 51 | ||
| 4 | + | 36 | = | 40 | ||
| 6 | + | 24 | = | 30 | ||
| 8 | + | 18 | = | 26 | ||
| 9 | + | 16 | = | 25 | ||
| 12 | + | 12 | = | 24 | That's it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, 12 and 12
16x4 + 12x2 + 12x2 + 9
Step-4 : Add up the first 2 terms, pulling out like factors :
4x2 • (4x2+3)
Add up the last 2 terms, pulling out common factors :
3 • (4x2+3)
Step-5 : Add up the four terms of step 4 :
(4x2+3) • (4x2+3)
Which is the desired factorization
Polynomial Roots Calculator :
6.8 Find roots (zeroes) of : F(x) = 4x2+3
See theory in step 6.2
In this case, the Leading Coefficient is 4 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,3
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 7.00 | ||||||
| -1 | 2 | -0.50 | 4.00 | ||||||
| -1 | 4 | -0.25 | 3.25 | ||||||
| -3 | 1 | -3.00 | 39.00 | ||||||
| -3 | 2 | -1.50 | 12.00 | ||||||
| -3 | 4 | -0.75 | 5.25 | ||||||
| 1 | 1 | 1.00 | 7.00 | ||||||
| 1 | 2 | 0.50 | 4.00 | ||||||
| 1 | 4 | 0.25 | 3.25 | ||||||
| 3 | 1 | 3.00 | 39.00 | ||||||
| 3 | 2 | 1.50 | 12.00 | ||||||
| 3 | 4 | 0.75 | 5.25 |
Polynomial Roots Calculator found no rational roots
Polynomial Roots Calculator :
6.9 Find roots (zeroes) of : F(x) = 4x2+3
See theory in step 6.2
In this case, the Leading Coefficient is 4 and the Trailing Constant is 3.
The factor(s) are:
of the Leading Coefficient : 1,2 ,4
of the Trailing Constant : 1 ,3
Let us test ....
| P | Q | P/Q | F(P/Q) | Divisor | |||||
|---|---|---|---|---|---|---|---|---|---|
| -1 | 1 | -1.00 | 7.00 | ||||||
| -1 | 2 | -0.50 | 4.00 | ||||||
| -1 | 4 | -0.25 | 3.25 | ||||||
| -3 | 1 | -3.00 | 39.00 | ||||||
| -3 | 2 | -1.50 | 12.00 | ||||||
| -3 | 4 | -0.75 | 5.25 | ||||||
| 1 | 1 | 1.00 | 7.00 | ||||||
| 1 | 2 | 0.50 | 4.00 | ||||||
| 1 | 4 | 0.25 | 3.25 | ||||||
| 3 | 1 | 3.00 | 39.00 | ||||||
| 3 | 2 | 1.50 | 12.00 | ||||||
| 3 | 4 | 0.75 | 5.25 |
Polynomial Roots Calculator found no rational roots
Multiplying Exponential Expressions :
6.10 Multiply (4x2+3) by (4x2+3)
The rule says : To multiply exponential expressions which have the same base, add up their exponents.
In our case, the common base is (4x2+3) and the exponents are :
1 , as (4x2+3) is the same number as (4x2+3)1
and 1 , as (4x2+3) is the same number as (4x2+3)1
The product is therefore, (4x2+3)(1+1) = (4x2+3)2
Equation at the end of step 6 :
(4x2 + 3)2
—————————— = 0
16
Step 7 :
When a fraction equals zero :
7.1 When a fraction equals zero ...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:
(4x2+3)2
———————— • 16 = 0 • 16
16
Now, on the left hand side, the 16 cancels out the denominator, while, on the right hand side, zero times anything is still zero.
The equation now takes the shape :
(4x2+3)2 = 0
Solving a Single Variable Equation :
7.2 Solve : (4x2+3)2 = 0
(4x2+3) 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 : 4x2+3 = 0
Subtract 3 from both sides of the equation :
4x2 = -3
Divide both sides of the equation by 4:
x2 = -3/4 = -0.750
When two things are equal, their square roots are equal. Taking the square root of the two sides of the equation we get:
x = ± √ -3/4
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Accordingly, √ -3/4 =
√ -1• 3/4 =
√ -1 •√ 3/4 =
i • √ 3/4
The equation has no real solutions. It has 2 imaginary, or complex solutions.
x= 0.0000 + 0.8660 i
x= 0.0000 - 0.8660 i
Supplement : Solving Quadratic Equation Directly
Solving 16x4+24x2+9 = 0 directly Earlier we factored this polynomial by splitting the middle term. let us now solve the equation by Completing The Square and by using the Quadratic Formula
Solving a Single Variable Equation :
Equations which are reducible to quadratic :
8.1 Solve 16x4+24x2+9 = 0
This equation is reducible to quadratic. What this means is that using a new variable, we can rewrite this equation as a quadratic equation Using w , such that w = x2 transforms the equation into :
16w2+24w+9 = 0
Solving this new equation using the quadratic formula we get one solution :
w = -0.75000010.5
Two solutions were found :
- x= 0.0000 - 0.8660 i
- x= 0.0000 + 0.8660 i
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