Step by step solution :
Step 1 :Equation in ~ the end of step 1 : ((0 - 2x2) - x) - 2 = 0
Step 2 :
Step 3 :Pulling out prefer terms :3.1 traction out like factors:-2x2 - x - 2=-1•(2x2 + x + 2)Trying to element by separating the center term
3.2Factoring 2x2 + x + 2 The first term is, 2x2 that coefficient is 2.The middle term is, +x that coefficient is 1.The last term, "the constant", is +2Step-1 : multiply the coefficient of the very first term by the constant 2•2=4Step-2 : uncover two components of 4 who sum amounts to the coefficient the the middle term, i m sorry is 1.
Observation : No 2 such factors can be found !! Conclusion : Trinomial have the right to not be factoredEquation in ~ the end of step 3 :
-2x2 - x - 2 = 0
Step 4 :Parabola, finding the Vertex:4.1Find the crest ofy = -2x2-x-2Parabolas have a highest possible or a lowest allude called the Vertex.Our parabola opens down and accordingly has a highest allude (AKA absolute maximum). We understand this even before plotting "y" because the coefficient that the first term,-2, is an adverse (smaller than zero).Each parabola has a vertical heat of symmetry that passes v its vertex. As such symmetry, the heat of the contrary would, because that example, pass through the midpoint of the 2 x-intercepts (roots or solutions) of the parabola. The is, if the parabola has indeed two genuine solutions.Parabolas have the right to model numerous real life situations, such together the height over ground, of an item thrown upward, after ~ some duration of time. The peak of the parabola can carry out us v information, such together the maximum height that object, thrown upwards, can reach. Thus we desire to have the ability to find the coordinates of the vertex.For any type of parabola,Ax2+Bx+C,the x-coordinate the the vertex is offered 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 - 2.0 or y = -1.875Parabola, Graphing Vertex and also X-Intercepts :
Root plot for : y = -2x2-x-2 Axis of the contrary (dashed) x=-0.25 Vertex at x,y = -0.25,-1.88 role has no real rootsSolve Quadratic Equation by completing The Square
4.2Solving-2x2-x-2 = 0 by completing The Square.Multiply both sides of the equation by (-1) to acquire positive coefficient because that the first term: 2x2+x+2 = 0Divide both sides of the equation by 2 to have 1 as the coefficient of the very first term :x2+(1/2)x+1 = 0Subtract 1 from both next of the equation :x2+(1/2)x = -1Now the clever bit: take the coefficient the x, i m sorry is 1/2, divide by two, giving 1/4, and also finally square it providing 1/16Add 1/16 come both sides of the equation :On the ideal hand side us have:-1+1/16or, (-1/1)+(1/16)The typical denominator of the 2 fractions is 16Adding (-16/16)+(1/16) provides -15/16So adding to both political parties we ultimately get:x2+(1/2)x+(1/16) = -15/16Adding 1/16 has completed the left hand side into a perfect square :x2+(1/2)x+(1/16)=(x+(1/4))•(x+(1/4))=(x+(1/4))2 things which room equal come the very same thing are additionally equal come one another. Sincex2+(1/2)x+(1/16) = -15/16 andx2+(1/2)x+(1/16) = (x+(1/4))2 then, according to the law of transitivity,(x+(1/4))2 = -15/16We"ll describe this Equation as Eq. #4.2.1 The Square root Principle claims that when two things are equal, their square roots are equal.Note the the square root of(x+(1/4))2 is(x+(1/4))2/2=(x+(1/4))1=x+(1/4)Now, applying the Square root Principle to Eq.#4.2.1 us get:x+(1/4)= √ -15/16 Subtract 1/4 from both sides to obtain:x = -1/4 + √ -15/16 In Math,iis called the imaginary unit. That satisfies i2=-1. Both i and also -i space the square root of -1Since a square root has two values, one positive and also the other negativex2 + (1/2)x + 1 = 0has two solutions:x = -1/4 + √ 15/16 • iorx = -1/4 - √ 15/16 • iNote that √ 15/16 can be created as√15 / √16which is √15 / 4
Solve Quadratic Equation utilizing the Quadratic Formula
4.3Solving-2x2-x-2 = 0 by the Quadratic Formula.According come the Quadratic Formula,x, the systems forAx2+Bx+C= 0 , whereby A, B and C room numbers, often dubbed coefficients, is provided by :-B± √B2-4ACx = ————————2A In our case,A= -2B= -1C= -2 Accordingly,B2-4AC=1 - 16 =-15Applying the quadratic formula : 1 ± √ -15 x=—————-4In the collection of real numbers, an unfavorable numbers carry out not have square roots.
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A new set the numbers, called complex, was developed so that negative numbers would have a square root. This numbers room written (a+b*i)Both i and -i room the square root of minus 1Accordingly,√-15=√15•(-1)=√15•√-1=±√ 15 •i √ 15 , rounded come 4 decimal digits, is 3.8730So now we are looking at:x=(1± 3.873 ns )/-4Two imaginary services :
x =(1+√-15)/-4=1/-4-i/4√ 15 = -0.2500+0.9682ior: x =(1-√-15)/-4=1/-4+i/4√ 15 = -0.2500-0.9682i