Step by step solution :

Step 1 :

Trying khổng lồ factor as a Difference of Cubes:1.1 Factoring: x3-8 Theory : A difference 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:8is the cube of 2Check: x3 is the cube of x1Factorization is :(x - 2)•(x2 + 2x + 4)

Trying to factor by splitting the middle term

1.2Factoring x2 + 2x + 4 The first term is, x2 its coefficient is 1.The middle term is, +2x its coefficient is 2.The last term, "the constant", is +4Step-1 : Multiply the coefficient of the first term by the constant 1•4=4Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is 2.

-4+-1=-5
-2+-2=-4
-1+-4=-5
1+4=5
2+2=4
4+1=5

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

Equation at the over of step 1 :

(x - 2) • (x2 + 2x + 4) = 0

Step 2 :

Theory - Roots of a hàng hóa :2.1 A product of several terms equals zero.When a product 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 sản phẩm = 0 as well.

Solving a Single Variable Equation:2.2Solve:x-2 = 0Add 2 khổng lồ both sides of the equation:x = 2

Parabola, Finding the Vertex:2.3Find the Vertex ofy = x2+2x+4Parabolas 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,1, 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 khổng lồ be able to 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 -1.0000Plugging into the parabola formula -1.0000 for x we can calculate the y-coordinate:y = 1.0 * -1.00 * -1.00 + 2.0 * -1.00 + 4.0 or y = 3.000

Parabola, Graphing Vertex & X-Intercepts :

Root plot for : y = x2+2x+4 Axis of Symmetry (dashed) x=-1.00 Vertex at x,y = -1.00, 3.00 Function has no real roots

Solve Quadratic Equation by Completing The Square

2.4Solvingx2+2x+4 = 0 by Completing The Square.Subtract 4 from both side of the equation :x2+2x = -4Now the clever bit: Take the coefficient of x, which is 2, divide by two, giving 1, & finally square it giving 1Add 1 to lớn both sides of the equation :On the right hand side we have:-4+1or, (-4/1)+(1/1)The common denominator of the two fractions is 1Adding (-4/1)+(1/1) gives -3/1So adding to lớn both sides we finally get:x2+2x+1 = -3Adding 1 has completed the left hand side into a perfect square :x2+2x+1=(x+1)•(x+1)=(x+1)2 Things which are equal khổng lồ the same thing are also equal to lớn one another. Sincex2+2x+1 = -3 andx2+2x+1 = (x+1)2 then, according to lớn the law of transitivity,(x+1)2 = -3We"ll refer khổng lồ this Equation as Eq. #2.4.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x+1)2 is(x+1)2/2=(x+1)1=x+1Now, applying the Square Root Principle lớn Eq.#2.4.1 we get:x+1= √ -3 Subtract 1 from both sides lớn obtain:x = -1 + √ -3 In Math,iis called the imaginary unit. It satisfies i2=-1. Both i & -i are the square roots of -1Since a square root has two values, one positive & the other negativex2 + 2x + 4 = 0has two solutions:x = -1 + √ 3 • iorx = -1 - √ 3 • i

Solve Quadratic Equation using the Quadratic Formula

2.5Solvingx2+2x+4 = 0 by the Quadratic Formula.According lớn 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= 1B= 2C= 4 Accordingly,B2-4AC=4 - 16 =-12Applying the quadratic formula : -2 ± √ -12 x=——————2In the set of real numbers, negative numbers vì chưng not have square roots. A new mix of numbers, called complex, was invented so that negative numbers would have a square root.


Bạn đang xem: How to solve the equation x^3 = 8 over complex numbers? could you give an explanation why x is not only 2


Xem thêm: Bộ Đề Kiểm Tra Hóa 1 Tiết Chương 2 Lớp 8 Chương 2 Năm 2020 Đề 1

These numbers are written (a+b*i)Both i và -i are the square roots of minus 1Accordingly,√-12=√12•(-1)=√12•√-1=±√ 12 •i Can √ 12 be simplified ?Yes!The prime factorization of 12is2•2•3 to 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).√ 12 =√2•2•3 =±2 •√ 3 √ 3 , rounded to 4 decimal digits, is 1.7321So now we are looking at:x=(-2±2• 1.732 i )/2Two imaginary solutions :

x =(-2+√-12)/2=-1+i√ 3 = -1.0000+1.7321ior: x =(-2-√-12)/2=-1-i√ 3 = -1.0000-1.7321i

Three solutions were found :

x =(-2-√-12)/2=-1-i√ 3 = -1.0000-1.7321ix =(-2+√-12)/2=-1+i√ 3 = -1.0000+1.7321ix = 2