## Step by step solution :

## Step 1 :

1.1 Evaluate : (3x-1)2 = 9x2-6x+1## Step 2 :

Pulling out like terms :2.1 Pull out like factors:9x2 - 6x - 15=3•(3x2 - 2x - 5)Trying to factor by splitting the middle term2.2Factoring 3x2 - 2x - 5 The first term is, 3x2 its coefficient is 3.The middle term is, -2x its coefficient is -2.The last term, "the constant", is -5Step-1 : Multiply the coefficient of the first term by the constant 3•-5=-15Step-2 : Find two factors of -15 whose sum equals the coefficient of the middle term, which is -2.

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-15 | + | 1 | = | -14 | ||

-5 | + | 3 | = | -2 | That"s it |

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step2above, -5 and 33x2 - 5x+3x - 5Step-4 : địa chỉ up the first 2 terms, pulling out lượt thích factors:x•(3x-5) địa chỉ up the last 2 terms, pulling out common factors:1•(3x-5) Step-5:Add up the four terms of step4:(x+1)•(3x-5)Which is the desired factorization

Equation at the kết thúc of step 2 :3 • (3x - 5) • (x + 1) = 0

## Step 3 :

Theory - Roots of a product :3.1 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 lớn solve as many equations as there are terms in the productAny solution of term = 0 solves sản phẩm = 0 as well.Equations which are never true:3.2Solve:3=0This equation has no solution. A a non-zero constant never equals zero.

Solving a Single Variable Equation:3.3Solve:3x-5 = 0Add 5 lớn both sides of the equation:3x = 5 Divide both sides of the equation by 3:x = 5/3 = 1.667

Solving a Single Variable Equation:3.4Solve:x+1 = 0Subtract 1 from both sides of the equation:x = -1

### Supplement : Solving Quadratic Equation Directly

Solving 3x2-2x-5 = 0 directly Earlier we factored this polynomial by splitting the middle term. Let us now solve the equation by Completing The Square và by using the Quadratic FormulaParabola, Finding the Vertex:4.1Find the Vertex ofy = 3x2-2x-5Parabolas 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 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 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 - 5.0 or y = -5.333

Parabola, Graphing Vertex and X-Intercepts :Root plot for : y = 3x2-2x-5 Axis of Symmetry (dashed) x= 0.33 Vertex at x,y = 0.33,-5.33 x-Intercepts (Roots) : Root 1 at x,y = -1.00, 0.00 Root 2 at x,y = 1.67, 0.00

Solve Quadratic Equation by Completing The Square4.2Solving3x2-2x-5 = 0 by Completing The Square.Divide both sides of the equation by 3 lớn have 1 as the coefficient of the first term :x2-(2/3)x-(5/3) = 0Add 5/3 lớn both side of the equation : x2-(2/3)x = 5/3Now 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 to both sides of the equation :On the right hand side we have:5/3+1/9The common denominator of the two fractions is 9Adding (15/9)+(1/9) gives 16/9So adding khổng lồ both sides we finally get:x2-(2/3)x+(1/9) = 16/9Adding 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 khổng lồ the same thing are also equal khổng lồ one another. Sincex2-(2/3)x+(1/9) = 16/9 andx2-(2/3)x+(1/9) = (x-(1/3))2 then, according to lớn the law of transitivity,(x-(1/3))2 = 16/9We"ll refer lớn this Equation as Eq.

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#4.2.1 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 lớn Eq.#4.2.1 we get:x-(1/3)= √ 16/9 showroom 1/3 lớn both sides khổng lồ obtain:x = 1/3 + √ 16/9 Since a square root has two values, one positive & the other negativex2 - (2/3)x - (5/3) = 0has two solutions:x = 1/3 + √ 16/9 orx = 1/3 - √ 16/9 note that √ 16/9 can be written as√16 / √9which is 4 / 3

### Solve Quadratic Equation using the Quadratic Formula

4.3Solving3x2-2x-5 = 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= 3B= -2C= -5 Accordingly,B2-4AC=4 - (-60) = 64Applying the quadratic formula : 2 ± √ 64 x=—————6Can √ 64 be simplified ?Yes!The prime factorization of 64is2•2•2•2•2•2 to lớn be able to remove something from under the radical, there have to lớn be 2 instances of it (because we are taking a square i.e. Second root).√ 64 =√2•2•2•2•2•2 =2•2•2•√ 1 =±8 •√ 1 =±8 So now we are looking at:x=(2±8)/6Two real solutions:x =(2+√64)/6=(1+4)/3= 1.667 or:x =(2-√64)/6=(1-4)/3= -1.000