Step by action solution :
Step 1 :Equation at the finish of step 1 : (2x2 - x) - 1 = 0
Step 2 :Trying to factor by splitting the center term2.1Factoring 2x2-x-1 The an initial term is, 2x2 that coefficient is 2.The middle term is, -x that coefficient is -1.The last term, "the constant", is -1Step-1 : multiply the coefficient that the an initial term through the continuous 2•-1=-2Step-2 : find two components of -2 who sum amounts to the coefficient the the center term, i m sorry is -1.
Step-3 : Rewrite the polynomial dividing the center term using the two determinants found in step2above, -2 and 12x2 - 2x+1x - 1Step-4 : include up the an initial 2 terms, pulling out like factors:2x•(x-1) include up the critical 2 terms, pulling out usual factors:1•(x-1) Step-5:Add increase the 4 terms of step4:(2x+1)•(x-1)Which is the preferred factorizationEquation at the finish of action 2 :
(x - 1) • (2x + 1) = 0
Step 3 :Theory - root of a product :3.1 A product of numerous terms equates to zero.When a product of 2 or much more terms equals zero, then at least one the the terms must be zero.We shall now solve each term = 0 separatelyIn various other words, we are going to fix as numerous equations as there room terms in the productAny equipment of ax = 0 solves product = 0 together well.
Solving a single Variable Equation:3.2Solve:x-1 = 0Add 1 to both sides of the equation:x = 1
Solving a solitary Variable Equation:3.3Solve:2x+1 = 0Subtract 1 native both sides of the equation:2x = -1 division both political parties of the equation by 2:x = -1/2 = -0.500
Supplement : fixing Quadratic Equation DirectlySolving 2x2-x-1 = 0 directly Earlier us factored this polynomial by splitting the center term. Let us now solve the equation by completing The Square and by using the Quadratic Formula
Parabola, detect the Vertex:4.1Find the crest ofy = 2x2-x-1Parabolas have actually a highest or a lowest point called the Vertex.Our parabola opens up up and as necessary has a lowest allude (AKA absolute minimum).We understand this even before plotting "y" because the coefficient that the first term,2, is positive (greater than zero).Each parabola has a vertical heat of symmetry the passes v its vertex. Therefore symmetry, the heat of the opposite would, because that example, pass v the midpoint of the 2 x-intercepts (roots or solutions) the the parabola. The is, if the parabola has indeed two real solutions.Parabolas can model plenty of real life situations, such together the height above ground, of things thrown upward, after some period of time. The peak of the parabola can provide us through information, such together the maximum height that object, thrown upwards, have the right to reach. Hence we desire to have the ability to find the coordinates of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate that the vertex is given by -B/(2A). In our instance the x coordinate is 0.2500Plugging into the parabola formula 0.2500 because that x we deserve to calculate the y-coordinate:y = 2.0 * 0.25 * 0.25 - 1.0 * 0.25 - 1.0 or y = -1.125Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = 2x2-x-1 Axis of symmetry (dashed) x= 0.25 Vertex in ~ x,y = 0.25,-1.12 x-Intercepts (Roots) : root 1 in ~ x,y = -0.50, 0.00 root 2 in ~ x,y = 1.00, 0.00Solve Quadratic Equation by completing The Square
4.2Solving2x2-x-1 = 0 by completing The Square.Divide both sides of the equation by 2 to have actually 1 together the coefficient of the very first term :x2-(1/2)x-(1/2) = 0Add 1/2 come both next of the equation : x2-(1/2)x = 1/2Now the clever bit: take it the coefficient that x, i beg your pardon is 1/2, division by two, offering 1/4, and finally square it giving 1/16Add 1/16 to both sides of the equation :On the ideal hand side us have:1/2+1/16The usual denominator of the two fractions is 16Adding (8/16)+(1/16) gives 9/16So adding to both political parties we ultimately get:x2-(1/2)x+(1/16) = 9/16Adding 1/16 has actually completed the left hand side right into a perfect square :x2-(1/2)x+(1/16)=(x-(1/4))•(x-(1/4))=(x-(1/4))2 points which room equal to the exact same thing are additionally equal come one another. Sincex2-(1/2)x+(1/16) = 9/16 andx2-(1/2)x+(1/16) = (x-(1/4))2 then, according to the legislation of transitivity,(x-(1/4))2 = 9/16We"ll describe this Equation together Eq.
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#4.2.1 The Square root Principle claims that once two things are equal, your square roots are equal.Note the the square source of(x-(1/4))2 is(x-(1/4))2/2=(x-(1/4))1=x-(1/4)Now, applying the Square source Principle come Eq.#4.2.1 we get:x-(1/4)= √ 9/16 include 1/4 to both sides to obtain:x = 1/4 + √ 9/16 since a square root has two values, one positive and also the other negativex2 - (1/2)x - (1/2) = 0has two solutions:x = 1/4 + √ 9/16 orx = 1/4 - √ 9/16 note that √ 9/16 can be composed as√9 / √16which is 3 / 4
Solve Quadratic Equation using the Quadratic Formula
4.3Solving2x2-x-1 = 0 by the Quadratic Formula.According come the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , wherein A, B and C are numbers, often referred to as coefficients, is provided by :-B± √B2-4ACx = ————————2A In ours case,A= 2B= -1C= -1 Accordingly,B2-4AC=1 - (-8) = 9Applying the quadratic formula : 1 ± √ 9 x=————4Can √ 9 be simplified ?Yes!The prime factorization of 9is3•3 To have the ability to remove something native under the radical, there have to be 2 instances of it (because we space taking a square i.e. 2nd root).√ 9 =√3•3 =±3 •√ 1 =±3 So currently we room looking at:x=(1±3)/4Two actual solutions:x =(1+√9)/4=(1+3)/4= 1.000 or:x =(1-√9)/4=(1-3)/4= -0.500