Rearrange:Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation : 3*x^2-2*x-1-(5)=0
Step by step solution :
Step 1 :Equation at the kết thúc of step 1 :
((3x2 - 2x) - 1) - 5 = 0
Step 2 :Trying khổng lồ factor by splitting the middle term2.1Factoring 3x2-2x-6 The first term is, 3x2 its coefficient is 3.The middle term is, -2x its coefficient is -2.The last term, "the constant", is -6Step-1 : Multiply the coefficient of the first term by the constant 3•-6=-18Step-2 : Find two factors of -18 whose sum equals the coefficient of the middle term, which is -2.
Observation : No two such factors can be found !! Conclusion : Trinomial can not be factoredEquation at the kết thúc of step 2 :
3x2 - 2x - 6 = 0
Step 3 :Parabola, Finding the Vertex:3.1Find the Vertex ofy = 3x2-2x-6Parabolas 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,3, 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 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 lớn 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 0.3333Plugging into the parabola formula 0.3333 for x we can calculate the y-coordinate:y = 3.0 * 0.33 * 0.33 - 2.0 * 0.33 - 6.0 or y = -6.333Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 3x2-2x-6 Axis of Symmetry (dashed) x= 0.33 Vertex at x,y = 0.33,-6.33 x-Intercepts (Roots) : Root 1 at x,y = -1.12, 0.00 Root 2 at x,y = 1.79, 0.00Solve Quadratic Equation by Completing The Square
3.2Solving3x2-2x-6 = 0 by Completing The Square.Divide both sides of the equation by 3 to have 1 as the coefficient of the first term :x2-(2/3)x-2 = 0Add 2 lớn both side of the equation : x2-(2/3)x = 2Now the clever bit: Take the coefficient of x, which is 2/3, divide by two, giving 1/3, và finally square it giving 1/9Add 1/9 khổng lồ both sides of the equation :On the right hand side we have:2+1/9or, (2/1)+(1/9)The common denominator of the two fractions is 9Adding (18/9)+(1/9) gives 19/9So adding lớn both sides we finally get:x2-(2/3)x+(1/9) = 19/9Adding 1/9 has completed the left hand side into a perfect square :x2-(2/3)x+(1/9)=(x-(1/3))•(x-(1/3))=(x-(1/3))2 Things which are equal to the same thing are also equal lớn one another. Sincex2-(2/3)x+(1/9) = 19/9 andx2-(2/3)x+(1/9) = (x-(1/3))2 then, according to the law of transitivity,(x-(1/3))2 = 19/9We"ll refer khổng lồ this Equation as Eq.
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#3.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that the square root of(x-(1/3))2 is(x-(1/3))2/2=(x-(1/3))1=x-(1/3)Now, applying the Square Root Principle to Eq.#3.2.1 we get:x-(1/3)= √ 19/9 showroom 1/3 khổng lồ both sides khổng lồ obtain:x = 1/3 + √ 19/9 Since a square root has two values, one positive and the other negativex2 - (2/3)x - 2 = 0has two solutions:x = 1/3 + √ 19/9 orx = 1/3 - √ 19/9 note that √ 19/9 can be written as√19 / √9which is √19 / 3
Solve Quadratic Equation using the Quadratic Formula
3.3Solving3x2-2x-6 = 0 by the Quadratic Formula.According khổng lồ the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B and C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 3B= -2C= -6 Accordingly,B2-4AC=4 - (-72) = 76Applying the quadratic formula : 2 ± √ 76 x=—————6Can √ 76 be simplified ?Yes!The prime factorization of 76is2•2•19 lớn be able khổng lồ remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. Second root).√ 76 =√2•2•19 =±2 •√ 19 √ 19 , rounded to lớn 4 decimal digits, is 4.3589So now we are looking at:x=(2±2• 4.359 )/6Two real solutions:x =(2+√76)/6=(1+√ 19 )/3= 1.786 or:x =(2-√76)/6=(1-√ 19 )/3= -1.120