This solution đơn hàng with finding the roots (zeroes) of polynomials.
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Step by Step Solution
Reformatting the input đầu vào :Changes made khổng lồ your input đầu vào should not affect the solution: (1): "x2" was replaced by "x^2". 1 more similar replacement(s).
Step by step solution :
Step 1 :Equation at the kết thúc of step 1 :
(2x3 + x2) - 9 = 0
Step 2 :
Polynomial Roots Calculator :2.1 Find roots (zeroes) of : F(x) = 2x3+x2-9Polynomial Roots Calculator is a set of methods aimed at finding values ofxfor which F(x)=0 Rational Roots chạy thử 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 p is a factor of the Trailing Constant & Q is a factor of the Leading CoefficientIn this case, the Leading Coefficient is 2 & the Trailing Constant is -9. The factor(s) are: of the Leading Coefficient : 1,2 of the Trailing Constant : 1 ,3 ,9 Let us demo ....
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p cảnh báo that q and phường originate from P/Q reduced khổng lồ its lowest terms In our case this means that 2x3+x2-9can be divided with 2x-3
Polynomial Long Division :
2.2 Polynomial Long Division Dividing : 2x3+x2-9("Dividend") By:2x-3("Divisor")
Quotient : x2+2x+3 Remainder: 0Trying lớn factor by splitting the middle term
2.3Factoring x2+2x+3 The first term is, x2 its coefficient is 1.The middle term is, +2x its coefficient is 2.The last term, "the constant", is +3Step-1 : Multiply the coefficient of the first term by the constant 1•3=3Step-2 : Find two factors of 3 whose sum equals the coefficient of the middle term, which is 2.
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Observation : No two such factors can be found !! Conclusion : Trinomial can not be factoredEquation at the end of step 2 :
(x2 + 2x + 3) • (2x - 3) = 0
Step 3 :Theory - Roots of a sản phẩm :3.1 A sản phẩm 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 khổng lồ 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:3.2Find the Vertex ofy = x2+2x+3Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up và 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 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 -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 + 3.0 or y = 2.000Parabola, Graphing Vertex & X-Intercepts :
Root plot for : y = x2+2x+3 Axis of Symmetry (dashed) x=-1.00 Vertex at x,y = -1.00, 2.00 Function has no real rootsSolve Quadratic Equation by Completing The Square
3.3Solvingx2+2x+3 = 0 by Completing The Square.Subtract 3 from both side of the equation :x2+2x = -3Now the clever bit: Take the coefficient of x, which is 2, divide by two, giving 1, và finally square it giving 1Add 1 to lớn both sides of the equation :On the right hand side we have:-3+1or, (-3/1)+(1/1)The common denominator of the two fractions is 1Adding (-3/1)+(1/1) gives -2/1So adding khổng lồ both sides we finally get:x2+2x+1 = -2Adding 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 to the same thing are also equal to lớn one another. Sincex2+2x+1 = -2 andx2+2x+1 = (x+1)2 then, according khổng lồ the law of transitivity,(x+1)2 = -2We"ll refer to lớn this Equation as Eq. #3.3.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x+1)2 is(x+1)2/2=(x+1)1=x+1Now, applying the Square Root Principle khổng lồ Eq.#3.3.1 we get:x+1= √ -2 Subtract 1 from both sides lớn obtain:x = -1 + √ -2 In Math,iis called the imaginary unit. It satisfies i2=-1. Both i and -i are the square roots of -1Since a square root has two values, one positive và the other negativex2 + 2x + 3 = 0has two solutions:x = -1 + √ 2 • iorx = -1 - √ 2 • i
Solve Quadratic Equation using the Quadratic Formula
3.4Solvingx2+2x+3 = 0 by the Quadratic Formula.According to 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= 3 Accordingly,B2-4AC=4 - 12 =-8Applying the quadratic formula : -2 ± √ -8 x=—————2In the set of real numbers, negative numbers vày not have square roots. A new set 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,√-8=√8•(-1)=√8•√-1=±√ 8 •i Can √ 8 be simplified ?Yes!The prime factorization of 8is2•2•2 lớn be able to 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).√ 8 =√2•2•2 =±2 •√ 2 √ 2 , rounded khổng lồ 4 decimal digits, is 1.4142So now we are looking at:x=(-2±2• 1.414 i )/2Two imaginary solutions :
x =(-2+√-8)/2=-1+i√ 2 = -1.0000+1.4142ior: x =(-2-√-8)/2=-1-i√ 2 = -1.0000-1.4142iSolving a Single Variable Equation:3.5Solve:2x-3 = 0Add 3 khổng lồ both sides of the equation:2x = 3 Divide both sides of the equation by 2:x = 3/2 = 1.500