Step by action solution :

Step 1 :

Equation at the finish of action 1 : ((22•3x2) - 4x) - 1 = 0

Step 2 :

Trying to factor by splitting the center term2.1Factoring 12x2-4x-1 The very first term is, 12x2 the coefficient is 12.The center term is, -4x the coefficient is -4.The critical term, "the constant", is -1Step-1 : multiply the coefficient the the very first term by the continuous 12•-1=-12Step-2 : discover two factors of -12 whose sum amounts to the coefficient of the center term, i m sorry is -4.

-12+1=-11
-6+2=-4That"s it

Step-3 : Rewrite the polynomial splitting the middle term utilizing the two determinants found in step2above, -6 and also 212x2 - 6x+2x - 1Step-4 : add up the first 2 terms, pulling out prefer factors:6x•(2x-1) add up the last 2 terms, pulling out usual factors:1•(2x-1) Step-5:Add up the 4 terms that step4:(6x+1)•(2x-1)Which is the preferred factorization

Equation in ~ the finish of action 2 :

(2x - 1) • (6x + 1) = 0

Step 3 :

Theory - root of a product :3.1 A product of several terms amounts to zero.When a product of 2 or much more terms equates to zero, then at the very least one of the terms must be zero.We shall now solve every term = 0 separatelyIn other words, we are going to resolve as numerous equations as there room terms in the productAny solution of ax = 0 solves product = 0 together well.

Solving a single Variable Equation:

3.2Solve:2x-1 = 0Add 1 come both political parties of the equation:2x = 1 divide both political parties of the equation by 2:x = 1/2 = 0.500

Solving a solitary Variable Equation:

3.3Solve:6x+1 = 0Subtract 1 indigenous both sides of the equation:6x = -1 divide both political parties of the equation by 6:x = -1/6 = -0.167

Supplement : addressing Quadratic Equation Directly

Solving 12x2-4x-1 = 0 straight Earlier we factored this polynomial by splitting the center term. Allow us currently solve the equation by perfect The Square and also by using the Quadratic Formula

Parabola, finding the Vertex:

4.1Find the crest ofy = 12x2-4x-1Parabolas have actually a highest possible or a lowest allude called the Vertex.Our parabola opens up and appropriately has a lowest point (AKA pure minimum).We understand this even prior to plotting "y" since the coefficient the the an initial term,12, is confident (greater 보다 zero).Each parabola has actually a vertical line of symmetry the passes through its vertex. As such symmetry, the line of the opposite would, for example, pass v the midpoint of the two x-intercepts (roots or solutions) of the parabola. The is, if the parabola has indeed two genuine solutions.Parabolas can model plenty of real life situations, such as the height above ground, of things thrown upward, after some period of time. The peak of the parabola can carry out us with information, such as the maximum height that object, thrown upwards, deserve to reach. Thus we want to have the ability to find the coordinates of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate the the peak is provided by -B/(2A). In our case the x name: coordinates is 0.1667Plugging right into the parabola formula 0.1667 for x we deserve to calculate the y-coordinate:y = 12.0 * 0.17 * 0.17 - 4.0 * 0.17 - 1.0 or y = -1.333

Parabola, Graphing Vertex and also X-Intercepts :

Root plot because that : y = 12x2-4x-1 Axis of the opposite (dashed) x= 0.17 Vertex in ~ x,y = 0.17,-1.33 x-Intercepts (Roots) : root 1 at x,y = -0.17, 0.00 source 2 in ~ x,y = 0.50, 0.00

Solve Quadratic Equation by completing The Square

4.2Solving12x2-4x-1 = 0 by perfect The Square.Divide both political parties of the equation through 12 to have actually 1 together the coefficient that the first term :x2-(1/3)x-(1/12) = 0Add 1/12 come both side of the equation : x2-(1/3)x = 1/12Now the clever bit: take it the coefficient the x, i m sorry is 1/3, divide by two, giving 1/6, and finally square it providing 1/36Add 1/36 come both sides of the equation :On the right hand side we have:1/12+1/36The common denominator the the two fractions is 36Adding (3/36)+(1/36) gives 4/36So including to both sides we ultimately get:x2-(1/3)x+(1/36) = 1/9Adding 1/36 has completed the left hand side right into a perfect square :x2-(1/3)x+(1/36)=(x-(1/6))•(x-(1/6))=(x-(1/6))2 points which space equal to the very same thing are likewise equal to one another. Sincex2-(1/3)x+(1/36) = 1/9 andx2-(1/3)x+(1/36) = (x-(1/6))2 then, according to the legislation of transitivity,(x-(1/6))2 = 1/9We"ll describe this Equation as Eq.


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#4.2.1 The Square root Principle says that when two things room equal, their square roots room equal.Note that the square root of(x-(1/6))2 is(x-(1/6))2/2=(x-(1/6))1=x-(1/6)Now, applying the Square root Principle come Eq.#4.2.1 us get:x-(1/6)= √ 1/9 add 1/6 come both sides to obtain:x = 1/6 + √ 1/9 because a square root has actually two values, one positive and the various other negativex2 - (1/3)x - (1/12) = 0has 2 solutions:x = 1/6 + √ 1/9 orx = 1/6 - √ 1/9 keep in mind that √ 1/9 can be composed as√1 / √9which is 1 / 3

Solve Quadratic Equation utilizing the Quadratic Formula

4.3Solving12x2-4x-1 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the systems forAx2+Bx+C= 0 , whereby A, B and C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 12B= -4C= -1 Accordingly,B2-4AC=16 - (-48) = 64Applying the quadratic formula : 4 ± √ 64 x=—————24Can √ 64 be simplified ?Yes!The prime factorization the 64is2•2•2•2•2•2 To have the ability to remove something from under the radical, there have to be 2 instances of that (because we space taking a square i.e. Second root).√ 64 =√2•2•2•2•2•2 =2•2•2•√ 1 =±8 •√ 1 =±8 So now we room looking at:x=(4±8)/24Two genuine solutions:x =(4+√64)/24=1/6+1/3= 0.500 or:x =(4-√64)/24=1/6-1/3= -0.167