7/2x-3=4/x

This deals with adding, subtracting and also finding the least usual multiple.

You are watching: Simplify 7(-2x - 3)


Step by step Solution

*

Rearrange:

Rearrange the equation by individually what is to the best of the equal sign from both political parties of the equation : 7/2*x-3-(4/x)=0

Step by action solution :

Step 1 :

4 leveling — xEquation at the end of action 1 : 7 4 ((— • x) - 3) - — = 0 2 x

Step 2 :

7 simplify — 2Equation at the end of action 2 : 7 4 ((— • x) - 3) - — = 0 2 x

Step 3 :

Rewriting the whole as an Equivalent fraction :3.1Subtracting a totality from a fraction Rewrite the whole as a fraction using 2 as the denominator :

3 3 • 2 3 = — = ————— 1 2 Equivalent portion : The fraction thus produced looks different but has the exact same value as the whole usual denominator : The equivalent fraction and the other fraction involved in the calculation share the exact same denominator

Adding fountain that have actually a common denominator :3.2 including up the two equivalent fractions add the two indistinguishable fractions i beg your pardon now have actually a usual denominatorCombine the numerators together, put the amount or difference over the common denominator then minimize to lowest terms if possible:

7x - (3 • 2) 7x - 6 ———————————— = —————— 2 2 Equation at the finish of action 3 : (7x - 6) 4 ———————— - — = 0 2 x

Step 4 :

Calculating the Least usual Multiple :4.1 uncover the Least usual Multiple The left denominator is : 2 The best denominator is : x

Number the times every prime factorappears in the administer of:PrimeFactorLeftDenominatorRightDenominatorL.C.M = MaxLeft,Right
2101
Product of allPrime Factors212

Number that times every Algebraic Factorappears in the factorization of:AlgebraicFactorLeftDenominatorRightDenominatorL.C.M = MaxLeft,Right
x011

Least usual Multiple: 2x

Calculating multiplier :

4.2 calculation multipliers because that the 2 fractions denote the Least typical Multiple through L.C.M signify the Left Multiplier through Left_M denote the appropriate Multiplier through Right_M denote the Left Deniminator through L_Deno represent the right Multiplier by R_DenoLeft_M=L.C.M/L_Deno=xRight_M=L.C.M/R_Deno=2

Making indistinguishable Fractions :

4.3 Rewrite the two fractions into indistinguishable fractionsTwo fountain are dubbed equivalent if they have the exact same numeric value. For example : 1/2 and also 2/4 room equivalent, y/(y+1)2 and also (y2+y)/(y+1)3 are equivalent as well. To calculate equivalent fraction , main point the molecule of every fraction, through its corresponding Multiplier.

L. Mult. • L. Num. (7x-6) • x —————————————————— = —————————— L.C.M 2x R. Mult. • R. Num. 4 • 2 —————————————————— = ————— L.C.M 2x including fractions that have actually a common denominator :4.4 adding up the two indistinguishable fractions

(7x-6) • x - (4 • 2) 7x2 - 6x - 8 ———————————————————— = ———————————— 2x 2x do the efforts to element by dividing the middle term4.5Factoring 7x2 - 6x - 8 The first term is, 7x2 the coefficient is 7.The center term is, -6x the coefficient is -6.The last term, "the constant", is -8Step-1 : main point the coefficient the the very first term through the continuous 7•-8=-56Step-2 : discover two factors of -56 who sum equates to the coefficient the the center term, i beg your pardon is -6.

-56+1=-55
-28+2=-26
-14+4=-10
-8+7=-1
-7+8=1
-4+14=10
-2+28=26
-1+56=55

Observation : No two such factors can be discovered !! Conclusion : Trinomial can not it is in factored

Equation at the finish of action 4 : 7x2 - 6x - 8 ———————————— = 0 2x

Step 5 :

When a fraction equals zero :5.1 as soon as a portion equals zero ...Where a portion equals zero, its numerator, the part which is over the portion line, must equal zero.Now,to get rid of the denominator, Tiger multiplys both political parties of the equation through the denominator.Here"s how:

7x2-6x-8 ———————— • 2x = 0 • 2x 2x Now, top top the left hand side, the 2x cancels the end the denominator, while, on the best hand side, zero times anything is tho zero.The equation now takes the shape:7x2-6x-8=0

Parabola, detect the Vertex:5.2Find the vertex ofy = 7x2-6x-8Parabolas have a highest possible or a lowest point called the Vertex.Our parabola opens up and appropriately has a lowest suggest (AKA pure minimum).We recognize this even before plotting "y" due to the fact that the coefficient the the very first term,7, is confident (greater than zero).Each parabola has a vertical line of symmetry that passes with its vertex. Because of this symmetry, the line of the opposite would, because that example, pass through the midpoint that the 2 x-intercepts (roots or solutions) of the parabola. The is, if the parabola has indeed two real solutions.Parabolas have the right to model numerous real life situations, such together the height above ground, of an object thrown upward, after ~ some duration of time. The crest of the parabola can administer us through information, such as the maximum elevation that object, thrown upwards, have the right to reach. Therefore we want to be able to find the works with of the vertex.For any kind of parabola,Ax2+Bx+C,the x-coordinate that the vertex is provided by -B/(2A). In our case the x coordinate is 0.4286Plugging into the parabola formula 0.4286 because that x we have the right to calculate the y-coordinate:y = 7.0 * 0.43 * 0.43 - 6.0 * 0.43 - 8.0 or y = -9.286

Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = 7x2-6x-8 Axis of the opposite (dashed) x= 0.43 Vertex in ~ x,y = 0.43,-9.29 x-Intercepts (Roots) : root 1 at x,y = -0.72, 0.00 source 2 in ~ x,y = 1.58, 0.00

Solve Quadratic Equation by perfect The Square

5.3Solving7x2-6x-8 = 0 by perfect The Square.Divide both sides of the equation by 7 to have actually 1 as the coefficient that the first term :x2-(6/7)x-(8/7) = 0Add 8/7 to both side of the equation : x2-(6/7)x = 8/7Now the clever bit: take the coefficient that x, i m sorry is 6/7, division by two, providing 3/7, and also finally square it offering 9/49Add 9/49 to both sides of the equation :On the appropriate hand side we have:8/7+9/49The common denominator of the 2 fractions is 49Adding (56/49)+(9/49) gives 65/49So adding to both sides we ultimately get:x2-(6/7)x+(9/49) = 65/49Adding 9/49 has completed the left hand side into a perfect square :x2-(6/7)x+(9/49)=(x-(3/7))•(x-(3/7))=(x-(3/7))2 points which are equal come the exact same thing are also equal come one another. Sincex2-(6/7)x+(9/49) = 65/49 andx2-(6/7)x+(9/49) = (x-(3/7))2 then, follow to the regulation of transitivity,(x-(3/7))2 = 65/49We"ll describe this Equation as Eq.

See more: Which Is Equal To The Ideal Mechanical Advantage Of A Pulley System

#5.3.1 The Square source Principle says that as soon as two things room equal, their square roots room equal.Note that the square root of(x-(3/7))2 is(x-(3/7))2/2=(x-(3/7))1=x-(3/7)Now, applying the Square source Principle to Eq.#5.3.1 we get:x-(3/7)= √ 65/49 add 3/7 come both sides to obtain:x = 3/7 + √ 65/49 since a square root has two values, one positive and the other negativex2 - (6/7)x - (8/7) = 0has 2 solutions:x = 3/7 + √ 65/49 orx = 3/7 - √ 65/49 keep in mind that √ 65/49 have the right to be written as√65 / √49which is √65 / 7

Solve Quadratic Equation using the Quadratic Formula

5.4Solving7x2-6x-8 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , whereby A, B and C space numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 7B= -6C= -8 Accordingly,B2-4AC=36 - (-224) = 260Applying the quadratic formula : 6 ± √ 260 x=—————14Can √ 260 be simplified ?Yes!The element factorization that 260is2•2•5•13 To be able to remove something native under the radical, there need to be 2 instances of the (because we room taking a square i.e. 2nd root).√ 260 =√2•2•5•13 =±2 •√ 65 √ 65 , rounded to 4 decimal digits, is 8.0623So now we space looking at:x=(6±2• 8.062 )/14Two actual solutions:x =(6+√260)/14=(3+√ 65 )/7= 1.580 or:x =(6-√260)/14=(3-√ 65 )/7= -0.723

Two options were discovered :

x =(6-√260)/14=(3-√ 65 )/7= -0.723 x =(6+√260)/14=(3+√ 65 )/7= 1.580