Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : (x-5)^3-(64)=0Step by step solution :
Step 1 :
1.1 Evaluate : (x-5)3 = x3-15x2+75x-125Checking for a perfect cube :1.2x3-15x2+75x-189 is not a perfect cube
Trying khổng lồ factor by pulling out :1.3 Factoring: x3-15x2+75x-189 Thoughtfully split the expression at hand into groups, each group having two terms:Group 1: x3-189Group 2: -15x2+75xPull out from each group separately :Group 1: (x3-189) • (1)Group 2: (x-5) • (-15x)Bad news !! Factoring by pulling out fails : The groups have no common factor and can not be added up to khung a multiplication.
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Polynomial Roots Calculator :
1.4 Find roots (zeroes) of : F(x) = x3-15x2+75x-189Polynomial Roots Calculator is a mix of methods aimed at finding values ofxfor which F(x)=0 Rational Roots thử nghiệm is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integersThe Rational Root Theorem states that if a polynomial zeroes for a rational numberP/Q then phường is a factor of the Trailing Constant and Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 1 và the Trailing Constant is -189. The factor(s) are: of the Leading Coefficient : 1of the Trailing Constant : 1 ,3 ,7 ,9 ,21 ,27 ,63 ,189 Let us chạy thử ....
| -1 | 1 | -1.00 | -280.00 | ||||||
| -3 | 1 | -3.00 | -576.00 | ||||||
| -7 | 1 | -7.00 | -1792.00 | ||||||
| -9 | 1 | -9.00 | -2808.00 | ||||||
| -21 | 1 | -21.00 | -17640.00 | ||||||
| -27 | 1 | -27.00 | -32832.00 | ||||||
| -63 | 1 | -63.00 | -314496.00 | ||||||
| -189 | 1 | -189.00 | -7301448.00 | ||||||
| 1 | 1 | 1.00 | -128.00 | ||||||
| 3 | 1 | 3.00 | -72.00 | ||||||
| 7 | 1 | 7.00 | -56.00 | ||||||
| 9 | 1 | 9.00 | 0.00 | x-9 | |||||
| 21 | 1 | 21.00 | 4032.00 | ||||||
| 27 | 1 | 27.00 | 10584.00 | ||||||
| 63 | 1 | 63.00 | 195048.00 | ||||||
| 189 | 1 | 189.00 | 6229440.00 |
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p lưu ý that q and p. Originate from P/Q reduced to lớn its lowest terms In our case this means that x3-15x2+75x-189can be divided with x-9
Polynomial Long Division :
1.5 Polynomial Long Division Dividing : x3-15x2+75x-189("Dividend") By:x-9("Divisor")
| dividend | x3 | - | 15x2 | + | 75x | - | 189 | ||
| -divisor | * x2 | x3 | - | 9x2 | |||||
| remainder | - | 6x2 | + | 75x | - | 189 | |||
| -divisor | * -6x1 | - | 6x2 | + | 54x | ||||
| remainder | 21x | - | 189 | ||||||
| -divisor | * 21x0 | 21x | - | 189 | |||||
| remainder | 0 |
Quotient : x2-6x+21 Remainder: 0
Trying lớn factor by splitting the middle term1.6Factoring x2-6x+21 The first term is, x2 its coefficient is 1.The middle term is, -6x its coefficient is -6.The last term, "the constant", is +21Step-1 : Multiply the coefficient of the first term by the constant 1•21=21Step-2 : Find two factors of 21 whose sum equals the coefficient of the middle term, which is -6.
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| -21 | + | -1 | = | -22 | ||
| -7 | + | -3 | = | -10 | ||
| -3 | + | -7 | = | -10 | ||
| -1 | + | -21 | = | -22 | ||
| 1 | + | 21 | = | 22 | ||
| 3 | + | 7 | = | 10 | ||
| 7 | + | 3 | = | 10 | ||
| 21 | + | 1 | = | 22 |
Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored
Equation at the kết thúc of step 1 :(x2 - 6x + 21) • (x - 9) = 0
Step 2 :
Theory - Roots of a sản phẩm :2.1 A hàng hóa 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 to solve as many equations as there are terms in the productAny solution of term = 0 solves hàng hóa = 0 as well.Parabola, Finding the Vertex:2.2Find the Vertex ofy = x2-6x+21Parabolas 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 to 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 3.0000Plugging into the parabola formula 3.0000 for x we can calculate the y-coordinate:y = 1.0 * 3.00 * 3.00 - 6.0 * 3.00 + 21.0 or y = 12.000
Parabola, Graphing Vertex và X-Intercepts :Root plot for : y = x2-6x+21 Axis of Symmetry (dashed) x= 3.00 Vertex at x,y = 3.00,12.00 Function has no real roots
Solve Quadratic Equation by Completing The Square2.3Solvingx2-6x+21 = 0 by Completing The Square.Subtract 21 from both side of the equation :x2-6x = -21Now the clever bit: Take the coefficient of x, which is 6, divide by two, giving 3, and finally square it giving 9Add 9 khổng lồ both sides of the equation :On the right hand side we have:-21+9or, (-21/1)+(9/1)The common denominator of the two fractions is 1Adding (-21/1)+(9/1) gives -12/1So adding to both sides we finally get:x2-6x+9 = -12Adding 9 has completed the left hand side into a perfect square :x2-6x+9=(x-3)•(x-3)=(x-3)2 Things which are equal lớn the same thing are also equal lớn one another. Sincex2-6x+9 = -12 andx2-6x+9 = (x-3)2 then, according to lớn the law of transitivity,(x-3)2 = -12We"ll refer lớn this Equation as Eq. #2.3.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-3)2 is(x-3)2/2=(x-3)1=x-3Now, applying the Square Root Principle lớn Eq.#2.3.1 we get:x-3= √ -12 add 3 khổng lồ both sides to lớn obtain:x = 3 + √ -12 In Math,iis called the imaginary unit. It satisfies i2=-1. Both i và -i are the square roots of -1Since a square root has two values, one positive và the other negativex2 - 6x + 21 = 0has two solutions:x = 3 + √ 12 • iorx = 3 - √ 12 • i
Solve Quadratic Equation using the Quadratic Formula
2.4Solvingx2-6x+21 = 0 by the Quadratic Formula.According to 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= -6C= 21 Accordingly,B2-4AC=36 - 84 =-48Applying the quadratic formula : 6 ± √ -48 x=—————2In the phối of real numbers, negative numbers bởi vì not have square roots. A new phối of numbers, called complex, was invented so that negative numbers would have a square root. These numbers are written (a+b*i)Both i & -i are the square roots of minus 1Accordingly,√-48=√48•(-1)=√48•√-1=±√ 48 •i Can √ 48 be simplified ?Yes!The prime factorization of 48is2•2•2•2•3 khổng lồ be able lớn remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. Second root).√ 48 =√2•2•2•2•3 =2•2•√ 3 =±4 •√ 3 √ 3 , rounded lớn 4 decimal digits, is 1.7321So now we are looking at:x=(6±4• 1.732 i )/2Two imaginary solutions :
x =(6+√-48)/2=3+2i√ 3 = 3.0000+3.4641ior: x =(6-√-48)/2=3-2i√ 3 = 3.0000-3.4641iSolving a Single Variable Equation:2.5Solve:x-9 = 0Add 9 lớn both sides of the equation:x = 9