## Step 1 :

Trying to aspect as a distinction of Cubes:1.1 Factoring: x3-1 concept : A distinction of two perfect cubes, a3-b3 can be factored into(a-b)•(a2+ab+b2)Proof:(a-b)•(a2+ab+b2)=a3+a2b+ab2-ba2-b2a-b3 =a3+(a2b-ba2)+(ab2-b2a)-b3=a3+0+0-b3=a3-b3Check:1is the cube that 1Check: x3 is the cube that x1Factorization is :(x - 1)•(x2 + x + 1)

Trying to aspect by separating the center term

1.2Factoring x2 + x + 1 The very first term is, x2 that is coefficient is 1.The center term is, +x that coefficient is 1.The critical term, "the constant", is +1Step-1 : main point the coefficient of the an initial term through the continuous 1•1=1Step-2 : find two components of 1 who sum equals the coefficient that the center term, i m sorry is 1.

 -1 + -1 = -2 1 + 1 = 2

Observation : No two such components can be uncovered !! Conclusion : Trinomial have the right to not it is in factored

Equation at the finish of action 1 :

(x - 1) • (x2 + x + 1) = 0

## Step 2 :

Theory - root of a product :2.1 A product of several terms equates to zero.When a product of two or much more terms equates to zero, climate at the very least one that the terms should be zero.We shall currently solve each term = 0 separatelyIn other words, we space going to fix as numerous equations as there room terms in the productAny equipment of hatchet = 0 solves product = 0 together well.

Solving a single Variable Equation:2.2Solve:x-1 = 0Add 1 come both sides of the equation:x = 1

Parabola, detect the Vertex:2.3Find the peak ofy = x2+x+1Parabolas have a highest or a lowest suggest called the Vertex.Our parabola opens up and appropriately has a lowest suggest (AKA pure minimum).We understand this even prior to plotting "y" because the coefficient the the very first term,1, is optimistic (greater 보다 zero).Each parabola has a vertical line of symmetry that passes through its vertex. As such symmetry, the heat of the opposite would, because that example, pass v the midpoint that the 2 x-intercepts (roots or solutions) the the parabola. That is, if the parabola has indeed two genuine solutions.Parabolas deserve to model many real life situations, such together the height over ground, of an item thrown upward, ~ some duration of time. The peak of the parabola can carry out us with information, such together the maximum height that object, thrown upwards, can reach. Thus we desire to be able to find the collaborates of the vertex.For any kind of parabola,Ax2+Bx+C,the x-coordinate of the peak is provided by -B/(2A). In our situation the x name: coordinates is -0.5000Plugging right into the parabola formula -0.5000 for x we deserve to calculate the y-coordinate:y = 1.0 * -0.50 * -0.50 + 1.0 * -0.50 + 1.0 or y = 0.750

Parabola, Graphing Vertex and X-Intercepts :

Root plot because that : y = x2+x+1 Axis of symmetry (dashed) x=-0.50 Vertex in ~ x,y = -0.50, 0.75 role has no genuine roots

Solve Quadratic Equation by perfect The Square

2.4Solvingx2+x+1 = 0 by completing The Square.Subtract 1 indigenous both side of the equation :x2+x = -1Now the clever bit: take it the coefficient of x, i m sorry is 1, division by two, providing 1/2, and finally square it providing 1/4Add 1/4 to both political parties of the equation :On the appropriate hand side us have:-1+1/4or, (-1/1)+(1/4)The common denominator the the two fractions is 4Adding (-4/4)+(1/4) offers -3/4So adding to both sides we ultimately get:x2+x+(1/4) = -3/4Adding 1/4 has completed the left hand side right into a perfect square :x2+x+(1/4)=(x+(1/2))•(x+(1/2))=(x+(1/2))2 things which space equal come the same thing are also equal come one another. Sincex2+x+(1/4) = -3/4 andx2+x+(1/4) = (x+(1/2))2 then, according to the legislation of transitivity,(x+(1/2))2 = -3/4We"ll refer to this Equation as Eq. #2.4.1 The Square root Principle states that once two things space equal, your square roots are equal.Note that the square source of(x+(1/2))2 is(x+(1/2))2/2=(x+(1/2))1=x+(1/2)Now, applying the Square source Principle come Eq.#2.4.1 us get:x+(1/2)= √ -3/4 Subtract 1/2 native both sides to obtain:x = -1/2 + √ -3/4 In Math,iis dubbed the imaginary unit. It satisfies i2=-1. Both i and also -i room the square root of -1Since a square root has two values, one positive and also the other negativex2 + x + 1 = 0has 2 solutions:x = -1/2 + √ 3/4 • iorx = -1/2 - √ 3/4 • iNote that √ 3/4 have the right to be written as√3 / √4which is √3 / 2

2.5Solvingx2+x+1 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the equipment forAx2+Bx+C= 0 , whereby A, B and also C are numbers, often called coefficients, is offered by :-B± √B2-4ACx = ————————2A In our case,A= 1B= 1C= 1 Accordingly,B2-4AC=1 - 4 =-3Applying the quadratic formula : -1 ± √ -3 x=—————2In the set of actual numbers, negative numbers do not have square roots.

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A new set that numbers, dubbed complex, was developed so that an unfavorable numbers would have actually a square root. This numbers are written (a+b*i)Both i and also -i room the square roots of minus 1Accordingly,√-3=√3•(-1)=√3•√-1=±√ 3 •i √ 3 , rounded come 4 decimal digits, is 1.7321So now we are looking at:x=(-1± 1.732 i )/2Two imaginary remedies :

x =(-1+√-3)/2=(-1+i√ 3 )/2= -0.5000+0.8660ior: x =(-1-√-3)/2=(-1-i√ 3 )/2= -0.5000-0.8660i

## Three services were discovered :

x =(-1-√-3)/2=(-1-i√ 3 )/2= -0.5000-0.8660ix =(-1+√-3)/2=(-1+i√ 3 )/2= -0.5000+0.8660ix = 1