1/3x^2-1/2x-1/3=0
This solution deals with adding, subtracting and finding the least common multiple.
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Step by Step Solution
Step by step solution :
Step 1 :
1 Simplify — 3Equation at the end of step 1 : 1 1 1 ((—•(x2))-(—•x))-— = 0 3 2 3Step 2 :
1 Simplify — 2Equation at the end of step 2 : 1 1 1 ((—•(x2))-(—•x))-— = 0 3 2 3Step 3 :
1 Simplify — 3Equation at the end of step 3 : 1 x 1 ((— • x2) - —) - — = 0 3 2 3Step 4 :
Equation at the over of step 4 : x2 x 1 (—— - —) - — = 0 3 2 3Step 5 :
Calculating the Least Common Multiple :5.1 Find the Least Common Multiple The left denominator is : 3 The right denominator is : 2| 3 | 1 | 0 | 1 |
| 2 | 0 | 1 | 1 |
| Product of allPrime Factors | 3 | 2 | 6 |
Least Common Multiple: 6
Calculating Multipliers :5.2 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_DenoLeft_M=L.C.M/L_Deno=2Right_M=L.C.M/R_Deno=3
Making Equivalent Fractions :5.3 Rewrite the two fractions into equivalent fractionsTwo fractions are called equivalent if they have the same numeric value. For example : 50% and 2/4 are equivalent, y/(y+1)2 & (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 • 2 —————————————————— = —————— L.C.M 6 R. Mult. • R. Num. X • 3 —————————————————— = ————— L.C.M 6 Adding fractions that have a common denominator :5.4 Adding up the two equivalent fractions add the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce lớn lowest terms if possible:
x2 • 2 - (x • 3) 2x2 - 3x ———————————————— = ———————— 6 6 Equation at the over of step 5 : (2x2 - 3x) 1 —————————— - — = 0 6 3
Step 6 :
Step 7 :
Pulling out like terms :7.1 Pull out like factors:2x2 - 3x=x•(2x - 3)Calculating the Least Common Multiple :7.2 Find the Least Common Multiple The left denominator is : 6 The right denominator is : 3
| 2 | 1 | 0 | 1 |
| 3 | 1 | 1 | 1 |
| Product of allPrime Factors | 6 | 3 | 6 |
Least Common Multiple: 6
Calculating Multipliers :7.3 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_DenoLeft_M=L.C.M/L_Deno=1Right_M=L.C.M/R_Deno=2
Making Equivalent Fractions :7.4 Rewrite the two fractions into equivalent fractions
L. Mult. • L. Num. X • (2x-3) —————————————————— = —————————— L.C.M 6 R. Mult. • R. Num. 2 —————————————————— = — L.C.M 6Adding fractions that have a common denominator :7.5 Adding up the two equivalent fractions
x • (2x-3) - (2) 2x2 - 3x - 2 ———————————————— = ———————————— 6 6 Trying lớn factor by splitting the middle term7.6Factoring 2x2 - 3x - 2 The first term is, 2x2 its coefficient is 2.The middle term is, -3x its coefficient is -3.The last term, "the constant", is -2Step-1 : Multiply the coefficient of the first term by the constant 2•-2=-4Step-2 : Find two factors of -4 whose sum equals the coefficient of the middle term, which is -3.
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| -4 | + | 1 | = | -3 | That"s it |
Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -4 & 12x2 - 4x+1x - 2Step-4 : add up the first 2 terms, pulling out lượt thích factors:2x•(x-2) địa chỉ cửa hàng up the last 2 terms, pulling out common factors:1•(x-2) Step-5:Add up the four terms of step4:(2x+1)•(x-2)Which is the desired factorization
Equation at the over of step 7 : (x - 2) • (2x + 1) —————————————————— = 0 6
Step 8 :
When a fraction equals zero :8.1 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:(x-2)•(2x+1) ———————————— • 6 = 0 • 6 6 Now, on the left hand side, the 6 cancels out the denominator, while, on the right hand side, zero times anything is still zero.The equation now takes the shape:(x-2) • (2x+1)=0
Theory - Roots of a sản phẩm :8.2 A hàng hóa of several terms equals zero.When a sản phẩm of two or more terms equals zero, then at least one of the terms must be zero.We shall now solve each term = 0 separatelyIn other words, we are going khổng lồ solve as many equations as there are terms in the productAny solution of term = 0 solves hàng hóa = 0 as well.
Solving a Single Variable Equation:8.3Solve:x-2 = 0Add 2 to lớn both sides of the equation:x = 2
Solving a Single Variable Equation:8.4Solve:2x+1 = 0Subtract 1 from both sides of the equation:2x = -1 Divide both sides of the equation by 2:x = -1/2 = -0.500
Supplement : Solving Quadratic Equation Directly
Solving 2x2-3x-2 = 0 directly Earlier we factored this polynomial by splitting the middle term. Let us now solve the equation by Completing The Square và by using the Quadratic FormulaParabola, Finding the Vertex:9.1Find the Vertex ofy = 2x2-3x-2Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up & accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,2, is positive (greater than zero).Each parabola has a vertical line of symmetry that passes through its vertex. Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x-intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.Parabolas can model many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. For this reason we want lớn be able to lớn find the coordinates of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate of the vertex is given by -B/(2A). In our case the x coordinate is 0.7500Plugging into the parabola formula 0.7500 for x we can calculate the y-coordinate:y = 2.0 * 0.75 * 0.75 - 3.0 * 0.75 - 2.0 or y = -3.125
Parabola, Graphing Vertex & X-Intercepts :Root plot for : y = 2x2-3x-2 Axis of Symmetry (dashed) x= 0.75 Vertex at x,y = 0.75,-3.12 x-Intercepts (Roots) : Root 1 at x,y = -0.50, 0.00 Root 2 at x,y = 2.00, 0.00
Solve Quadratic Equation by Completing The Square9.2Solving2x2-3x-2 = 0 by Completing The Square.Divide both sides of the equation by 2 to have 1 as the coefficient of the first term :x2-(3/2)x-1 = 0Add 1 khổng lồ both side of the equation : x2-(3/2)x = 1Now the clever bit: Take the coefficient of x, which is 3/2, divide by two, giving 3/4, và finally square it giving 9/16Add 9/16 khổng lồ both sides of the equation :On the right hand side we have:1+9/16or, (1/1)+(9/16)The common denominator of the two fractions is 16Adding (16/16)+(9/16) gives 25/16So adding khổng lồ both sides we finally get:x2-(3/2)x+(9/16) = 25/16Adding 9/16 has completed the left hand side into a perfect square :x2-(3/2)x+(9/16)=(x-(3/4))•(x-(3/4))=(x-(3/4))2 Things which are equal to lớn the same thing are also equal lớn one another. Sincex2-(3/2)x+(9/16) = 25/16 andx2-(3/2)x+(9/16) = (x-(3/4))2 then, according khổng lồ the law of transitivity,(x-(3/4))2 = 25/16We"ll refer khổng lồ this Equation as Eq. #9.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-(3/4))2 is(x-(3/4))2/2=(x-(3/4))1=x-(3/4)Now, applying the Square Root Principle khổng lồ Eq.#9.2.1 we get:x-(3/4)= √ 25/16 showroom 3/4 khổng lồ both sides to lớn obtain:x = 3/4 + √ 25/16 Since a square root has two values, one positive & the other negativex2 - (3/2)x - 1 = 0has two solutions:x = 3/4 + √ 25/16 orx = ba phần tư - √ 25/16 note that √ 25/16 can be written as√25 / √16which is 5 / 4