Reformatting the input :

Changes made to her input must not influence the solution: (1): "x2" was changed by "x^2".

You are watching: (x-2)(x-2)

Step by action solution :

Step 1 :

Trying to variable by dividing the middle term

1.1Factoring x2-x-2 The very first term is, x2 the coefficient is 1.The middle term is, -x the coefficient is -1.The last term, "the constant", is -2Step-1 : main point the coefficient that the very first term through the continuous 1•-2=-2Step-2 : uncover two determinants of -2 whose sum equals the coefficient of the center term, which is -1.

-2+1=-1That"s it

Step-3 : Rewrite the polynomial separating the center term making use of the two determinants found in step2above, -2 and 1x2 - 2x+1x - 2Step-4 : add up the an initial 2 terms, pulling out favor factors:x•(x-2) add up the last 2 terms, pulling out usual factors:1•(x-2) Step-5:Add increase the 4 terms of step4:(x+1)•(x-2)Which is the desired factorization

Equation at the end of action 1 :

(x + 1) • (x - 2) = 0

Step 2 :

Theory - root of a product :2.1 A product of several terms amounts to zero.When a product of two or much more terms equals zero, then at the very least one that the terms should be zero.We shall currently solve each term = 0 separatelyIn various other words, we space going to solve as numerous equations together there space terms in the productAny solution of ax = 0 solves product = 0 as well.

Solving a solitary Variable Equation:2.2Solve:x+1 = 0Subtract 1 native both political parties of the equation:x = -1

Solving a single Variable Equation:2.3Solve:x-2 = 0Add 2 come both sides of the equation:x = 2

Supplement : resolving Quadratic Equation Directly

Solving x2-x-2 = 0 straight Earlier us factored this polynomial by splitting the middle term. Allow us now solve the equation by completing The Square and by making use of the Quadratic Formula

Parabola, detect the Vertex:3.1Find the peak ofy = x2-x-2Parabolas have a highest possible or a lowest suggest called the Vertex.Our parabola opens up up and appropriately has a lowest allude (AKA pure minimum).We recognize this even before plotting "y" since the coefficient of the very first term,1, is hopeful (greater than zero).Each parabola has actually a vertical heat of symmetry the passes through its vertex. Therefore symmetry, the heat of the contrary would, for example, pass through the midpoint the the two x-intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.Parabolas have the right to model plenty of real life situations, such as the height over ground, of an item thrown upward, after ~ some duration of time. The crest of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach. Thus we desire to have the ability to find the works with of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate that the vertex is offered by -B/(2A). In our situation the x coordinate is 0.5000Plugging into the parabola formula 0.5000 for x we deserve to calculate the y-coordinate:y = 1.0 * 0.50 * 0.50 - 1.0 * 0.50 - 2.0 or y = -2.250

Parabola, Graphing Vertex and X-Intercepts :

Root plot because that : y = x2-x-2 Axis of the contrary (dashed) x= 0.50 Vertex at x,y = 0.50,-2.25 x-Intercepts (Roots) : root 1 at x,y = -1.00, 0.00 root 2 at x,y = 2.00, 0.00

Solve Quadratic Equation by perfect The Square

3.2Solvingx2-x-2 = 0 by completing The Square.Add 2 come both next of the equation : x2-x = 2Now the clever bit: take it the coefficient the x, which is 1, division by two, providing 1/2, and finally square it giving 1/4Add 1/4 to both political parties of the equation :On the appropriate hand side we have:2+1/4or, (2/1)+(1/4)The typical denominator the the 2 fractions is 4Adding (8/4)+(1/4) provides 9/4So including to both political parties we ultimately get:x2-x+(1/4) = 9/4Adding 1/4 has actually completed the left hand side right into a perfect square :x2-x+(1/4)=(x-(1/2))•(x-(1/2))=(x-(1/2))2 points which room equal come the exact same thing are additionally equal come one another. Sincex2-x+(1/4) = 9/4 andx2-x+(1/4) = (x-(1/2))2 then, according to the law of transitivity,(x-(1/2))2 = 9/4We"ll refer to this Equation together Eq.

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#3.2.1 The Square source Principle says that once two things space equal, their square roots room equal.Note the the square root of(x-(1/2))2 is(x-(1/2))2/2=(x-(1/2))1=x-(1/2)Now, applying the Square source Principle come Eq.#3.2.1 us get:x-(1/2)= √ 9/4 add 1/2 to both sides to obtain:x = 1/2 + √ 9/4 because a square root has two values, one positive and also the various other negativex2 - x - 2 = 0has 2 solutions:x = 1/2 + √ 9/4 orx = 1/2 - √ 9/4 keep in mind that √ 9/4 have the right to be composed as√9 / √4which is 3 / 2

Solve Quadratic Equation making use of the Quadratic Formula

3.3Solvingx2-x-2 = 0 by the Quadratic Formula.According come the Quadratic Formula,x, the equipment forAx2+Bx+C= 0 , where A, B and C room numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 1B= -1C= -2 Accordingly,B2-4AC=1 - (-8) = 9Applying the quadratic formula : 1 ± √ 9 x=————2Can √ 9 be streamlined ?Yes!The prime factorization that 9is3•3 To be able to remove something native under the radical, there need to be 2 instances of that (because we room taking a square i.e. Second root).√ 9 =√3•3 =±3 •√ 1 =±3 So now we space looking at:x=(1±3)/2Two actual solutions:x =(1+√9)/2=(1+3)/2= 2.000 or:x =(1-√9)/2=(1-3)/2= -1.000