### Reformatting the input :

Changes do to her input must not affect the solution: (1): "x2" was replaced by "x^2".

## Step 1 :

Equation at the end of action 1 :

(3x2 - 10x) - 2 = 0

## Step 2 :

Trying to variable by splitting the center term2.1Factoring 3x2-10x-2 The first term is, 3x2 the coefficient is 3.The center term is, -10x that coefficient is -10.The last term, "the constant", is -2Step-1 : multiply the coefficient that the an initial term by the constant 3•-2=-6Step-2 : uncover two determinants of -6 whose sum amounts to the coefficient that the center term, i beg your pardon is -10.

 -6 + 1 = -5 -3 + 2 = -1 -2 + 3 = 1 -1 + 6 = 5

Observation : No 2 such determinants can be uncovered !! Conclusion : Trinomial can not be factored

Equation at the end of step 2 :

3x2 - 10x - 2 = 0

## Step 3 :

Parabola, recognize the Vertex:3.1Find the crest ofy = 3x2-10x-2Parabolas have actually a greatest or a lowest point called the Vertex.Our parabola opens up and accordingly has a lowest point (AKA absolute minimum).We recognize this even before plotting "y" due to the fact that the coefficient the the an initial term,3, is confident (greater than zero).Each parabola has a vertical heat of symmetry the passes v its vertex. Thus symmetry, the heat of the opposite would, because that example, pass v the midpoint of the two x-intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two actual solutions.Parabolas can model numerous real life situations, such together the height over ground, of things thrown upward, after some period of time. The crest of the parabola can administer us with information, such as the maximum height that object, thrown upwards, deserve to reach. Hence we want to have the ability to find the collaborates of the vertex.For any type of parabola,Ax2+Bx+C,the x-coordinate that the vertex is given by -B/(2A). In our instance the x name: coordinates is 1.6667Plugging right into the parabola formula 1.6667 for x we deserve to calculate the y-coordinate:y = 3.0 * 1.67 * 1.67 - 10.0 * 1.67 - 2.0 or y = -10.333

Parabola, Graphing Vertex and also X-Intercepts :

Root plot because that : y = 3x2-10x-2 Axis of symmetry (dashed) x= 1.67 Vertex at x,y = 1.67,-10.33 x-Intercepts (Roots) : root 1 at x,y = -0.19, 0.00 root 2 at x,y = 3.52, 0.00

Solve Quadratic Equation by perfect The Square

3.2Solving3x2-10x-2 = 0 by perfect The Square.Divide both sides of the equation through 3 to have 1 as the coefficient the the an initial term :x2-(10/3)x-(2/3) = 0Add 2/3 to both next of the equation : x2-(10/3)x = 2/3Now the clever bit: take it the coefficient that x, which is 10/3, divide by two, giving 5/3, and finally square it providing 25/9Add 25/9 to both sides of the equation :On the ideal hand side we have:2/3+25/9The common denominator the the two fractions is 9Adding (6/9)+(25/9) provides 31/9So including to both sides we ultimately get:x2-(10/3)x+(25/9) = 31/9Adding 25/9 has completed the left hand side into a perfect square :x2-(10/3)x+(25/9)=(x-(5/3))•(x-(5/3))=(x-(5/3))2 things which space equal come the exact same thing are also equal come one another. Sincex2-(10/3)x+(25/9) = 31/9 andx2-(10/3)x+(25/9) = (x-(5/3))2 then, follow to the law of transitivity,(x-(5/3))2 = 31/9We"ll describe this Equation together Eq.

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#3.2.1 The Square root Principle claims that when two things room equal, their square roots space equal.Note the the square source of(x-(5/3))2 is(x-(5/3))2/2=(x-(5/3))1=x-(5/3)Now, applying the Square source Principle come Eq.#3.2.1 us get:x-(5/3)= √ 31/9 include 5/3 come both political parties to obtain:x = 5/3 + √ 31/9 since a square root has actually two values, one positive and also the various other negativex2 - (10/3)x - (2/3) = 0has 2 solutions:x = 5/3 + √ 31/9 orx = 5/3 - √ 31/9 note that √ 31/9 deserve to be composed as√31 / √9which is √31 / 3