Step by step solution :
Step 1 :
Equation at the end of step 1 : 23x3 - 1 = 0Step 2 :
Trying lớn factor as a Difference of Cubes:2.1 Factoring: 8x3-1 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: 8 is the cube of 2Check:1is the cube of 1Check: x3 is the cube of x1Factorization is :(2x - 1)•(4x2 + 2x + 1)Trying to lớn factor by splitting the middle term2.2Factoring 4x2 + 2x + 1 The first term is, 4x2 its coefficient is 4.The middle term is, +2x its coefficient is 2.The last term, "the constant", is +1Step-1 : Multiply the coefficient of the first term by the constant 4•1=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 kết thúc of step 2 :(2x - 1) • (4x2 + 2x + 1) = 0
Step 3 :
Theory - Roots of a sản phẩm :3.1 A hàng hóa of several terms equals zero.When a hàng hóa 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 to solve sầu as many equations as there are terms in the productAny solution of term = 0 solves product = 0 as well.Solving a Single Variable Equation:3.2Solve:2x-1 = 0Add 1 to both sides of the equation:2x = 1 Divide both sides of the equation by 2:x = 50% = 0.500
Parabola, Finding the Vertex:3.3Find the Vertex ofy = 4x2+2x+1Parabolas 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,4, is positive (greater than zero).Each parabola has a vertical line of symmetry that passes through its vertex.
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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 mã sản phẩm 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 to 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 -0.2500Plugging inkhổng lồ the parabola formula -0.2500 for x we can calculate the y-coordinate:y = 4.0 * -0.25 * -0.25 + 2.0 * -0.25 + 1.0 or y = 0.750Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = 4x2+2x+1 Axis of Symmetry (dashed) x=-0.25 Vertex at x,y = -0.25, 0.75 Function has no real roots
Solve sầu Quadratic Equation by Completing The Square3.4Solving4x2+2x+1 = 0 by Completing The Square.Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :x2+(1/2)x+(1/4) = 0Subtract 1/4 from both side of the equation :x2+(1/2)x = -1/4Now the clever bit: Take the coefficient of x, which is 50%, divide by two, giving 1/4, và finally square it giving 1/16Add 1/16 to lớn both sides of the equation :On the right hvà side we have:-1/4+1/16The common denominator of the two fractions is 16Adding (-4/16)+(1/16) gives -3/16So adding khổng lồ both sides we finally get:x2+(1/2)x+(1/16) = -3/16Adding 1/16 has completed the left hand side inkhổng lồ a perfect square :x2+(1/2)x+(1/16)=(x+(1/4))•(x+(1/4))=(x+(1/4))2 Things which are equal khổng lồ the same thing are also equal khổng lồ one another.
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Sincex2+(1/2)x+(1/16) = -3/16 andx2+(1/2)x+(1/16) = (x+(1/4))2 then, according to the law of transitivity,(x+(1/4))2 = -3/16We"ll refer khổng lồ this Equation as Eq. #3.4.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x+(1/4))2 is(x+(1/4))2/2=(x+(1/4))1=x+(1/4)Now, applying the Square Root Principle to Eq.#3.4.1 we get:x+(1/4)= √ -3/16 Subtract 1/4 from both sides lớn obtain:x = -1/4 + √ -3/16 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 sầu and the other negativex2 + (1/2)x + (1/4) = 0has two solutions:x = -1/4 + √ 3/16 • iorx = -1/4 - √ 3/16 • icảnh báo that √ 3/16 can be written as√3 / √16which is √3 / 4Solve sầu Quadratic Equation using the Quadratic Formula
3.5Solving4x2+2x+1 = 0 by the Quadratic Formula.According lớn the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B và C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 4B= 2C= 1 Accordingly,B2-4AC=4 - 16 =-12Applying the quadratic formula : -2 ± √ -12 x=——————8In the set of real numbers, negative numbers bởi not have square roots. A new mix of numbers, called complex, was invented so that negative numbers would have sầu a square root. These numbers are written (a+b*i)Both i and -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 sầu something from under the radical, there have to lớn be 2 instances of it (because we are taking a square i.e. second root).√ 12 =√2•2•3 =±2 •√ 3 √ 3 , rounded to lớn 4 decimal digits, is 1.7321So now we are looking at:x=(-2±2• 1.732 i )/8Two imaginary solutions :
x =(-2+√-12)/8=(-1+i√ 3 )/4= -0.2500+0.4330ior: x =(-2-√-12)/8=(-1-i√ 3 )/4= -0.2500-0.4330i
