Reformatting the input đầu vào :

Changes made khổng lồ your input đầu vào should not affect the solution: (1): ".3" was replaced by "(3/10)".

Step by step solution :

Step 1 :

3 Simplify —— 10Equation at the kết thúc of step 1 : 3 ((—— • x2) - 2x) - 3 = 0 10

Step 2 :

Equation at the kết thúc of step 2 : 3x2 (——— - 2x) - 3 = 0 10

Step 3 :

Rewriting the whole as an Equivalent Fraction :3.1Subtracting a whole from a fraction Rewrite the whole as a fraction using 10 as the denominator :

2x 2x • 10 2x = —— = ——————— 1 10 Equivalent fraction : The fraction thus generated looks different but has the same value as the whole Common denominator : The equivalent fraction và the other fraction involved in the calculation mô tả the same denominator

Adding fractions that have a common denominator :3.2 Adding up the two equivalent fractions địa chỉ cửa hàng the two equivalent fractions which now have a common denominatorCombine the numerators together, put the sum or difference over the common denominator then reduce to lowest terms if possible:

3x2 - (2x • 10) 3x2 - 20x ——————————————— = ————————— 10 10 Equation at the kết thúc of step 3 : (3x2 - 20x) ——————————— - 3 = 0 10

Step 4 :

Rewriting the whole as an Equivalent Fraction :4.1Subtracting a whole from a fraction Rewrite the whole as a fraction using 10 as the denominator :

3 3 • 10 3 = — = —————— 1 10

Step 5 :

Pulling out like terms :5.1 Pull out like factors:3x2 - 20x=x•(3x - 20)

Adding fractions that have a common denominator :5.2 Adding up the two equivalent fractions

x • (3x-20) - (3 • 10) 3x2 - 20x - 30 —————————————————————— = —————————————— 10 10 Trying to lớn factor by splitting the middle term5.3Factoring 3x2 - 20x - 30 The first term is, 3x2 its coefficient is 3.The middle term is, -20x its coefficient is -20.The last term, "the constant", is -30Step-1 : Multiply the coefficient of the first term by the constant 3•-30=-90Step-2 : Find two factors of -90 whose sum equals the coefficient of the middle term, which is -20.

-90+1=-89
-45+2=-43
-30+3=-27
-18+5=-13
-15+6=-9
-10+9=-1
-9+10=1
-6+15=9
-5+18=13
-3+30=27
-2+45=43
-1+90=89

Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored

Equation at the end of step 5 : 3x2 - 20x - 30 —————————————— = 0 10

Step 6 :

When a fraction equals zero :6.1 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:

3x2-20x-30 —————————— • 10 = 0 • 10 10 Now, on the left hand side, the 10 cancels out the denominator, while, on the right hand side, zero times anything is still zero.The equation now takes the shape:3x2-20x-30=0

Parabola, Finding the Vertex:6.2Find the Vertex ofy = 3x2-20x-30Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up and accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,3, 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 khổng lồ 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 3.3333Plugging into the parabola formula 3.3333 for x we can calculate the y-coordinate:y = 3.0 * 3.33 * 3.33 - 20.0 * 3.33 - 30.0 or y = -63.333

Parabola, Graphing Vertex và X-Intercepts :Root plot for : y = 3x2-20x-30 Axis of Symmetry (dashed) x= 3.33 Vertex at x,y = 3.33,-63.33 x-Intercepts (Roots) : Root 1 at x,y = -1.26, 0.00 Root 2 at x,y = 7.93, 0.00

Solve Quadratic Equation by Completing The Square

6.3Solving3x2-20x-30 = 0 by Completing The Square.Divide both sides of the equation by 3 khổng lồ have 1 as the coefficient of the first term :x2-(20/3)x-10 = 0Add 10 lớn both side of the equation : x2-(20/3)x = 10Now the clever bit: Take the coefficient of x, which is 20/3, divide by two, giving 10/3, và finally square it giving 100/9Add 100/9 to both sides of the equation :On the right hand side we have:10+100/9or, (10/1)+(100/9)The common denominator of the two fractions is 9Adding (90/9)+(100/9) gives 190/9So adding lớn both sides we finally get:x2-(20/3)x+(100/9) = 190/9Adding 100/9 has completed the left hand side into a perfect square :x2-(20/3)x+(100/9)=(x-(10/3))•(x-(10/3))=(x-(10/3))2 Things which are equal to lớn the same thing are also equal khổng lồ one another. Sincex2-(20/3)x+(100/9) = 190/9 andx2-(20/3)x+(100/9) = (x-(10/3))2 then, according lớn the law of transitivity,(x-(10/3))2 = 190/9We"ll refer to this Equation as Eq.


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#6.3.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-(10/3))2 is(x-(10/3))2/2=(x-(10/3))1=x-(10/3)Now, applying the Square Root Principle lớn Eq.#6.3.1 we get:x-(10/3)= √ 190/9 add 10/3 to both sides to obtain:x = 10/3 + √ 190/9 Since a square root has two values, one positive và the other negativex2 - (20/3)x - 10 = 0has two solutions:x = 10/3 + √ 190/9 orx = 10/3 - √ 190/9 chú ý that √ 190/9 can be written as√190 / √9which is √190 / 3

Solve Quadratic Equation using the Quadratic Formula

6.4Solving3x2-20x-30 = 0 by the Quadratic Formula.According khổng lồ the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B & C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 3B=-20C=-30 Accordingly,B2-4AC=400 - (-360) = 760Applying the quadratic formula : đôi mươi ± √ 760 x=——————6Can √ 760 be simplified ?Yes!The prime factorization of 760is2•2•2•5•19 lớn be able to remove something from under the radical, there have khổng lồ be 2 instances of it (because we are taking a square i.e. Second root).√ 760 =√2•2•2•5•19 =±2 •√ 190 √ 190 , rounded lớn 4 decimal digits, is 13.7840So now we are looking at:x=(20±2• 13.784 )/6Two real solutions:x =(20+√760)/6=(10+√ 190 )/3= 7.928 or:x =(20-√760)/6=(10-√ 190 )/3= -1.261

Two solutions were found :

x =(20-√760)/6=(10-√ 190 )/3= -1.261 x =(20+√760)/6=(10+√ 190 )/3= 7.928