What Is The Discriminant Of X2+X+1=0 And What Does That Mean?

Reformatting the input đầu vào :

Changes made to your input đầu vào should not affect the solution: (1): "x2" was replaced by "x^2".

Step by step solution :

Step 1 :

Trying to lớn factor by splitting the middle term

1.1Factoring x2-x-1 The first term is, x2 its coefficient is 1.The middle term is, -x its coefficient is -1.The last term, "the constant", is -1Step-1 : Multiply the coefficient of the first term by the constant 1•-1=-1Step-2 : Find two factors of -1 whose sum equals the coefficient of the middle term, which is -1.


Observation : No two such factors can be found !! Conclusion : Trinomial can not be factored

Equation at the over of step 1 :

x2 - x - 1 = 0

Step 2 :

Parabola, Finding the Vertex:2.1Find the Vertex ofy = x2-x-1Parabolas have a highest or a lowest point called the Vertex.Our parabola opens up & accordingly has a lowest point (AKA absolute minimum).We know this even before plotting "y" because the coefficient of the first term,1, is positive sầu (greater than zero).Each parabola has a vertical line of symmetry that passes through its vertex.

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Because of this symmetry, the line of symmetry would, for example, pass through the midpoint of the two x-intercepts (roots or solutions) of the parabola. That is, if the parabola has indeed two real solutions.Parabolas can Mã Sản Phẩm many real life situations, such as the height above ground, of an object thrown upward, after some period of time. The vertex of the parabola can provide us with information, such as the maximum height that object, thrown upwards, can reach.

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For this reason we want lớn be able khổng lồ find the coordinates of the vertex.For any parabola,Ax2+Bx+C,the x-coordinate of the vertex is given by -B/(2A). In our case the x coordinate is 0.5000Plugging into the parabola formula 0.5000 for x we can calculate the y-coordinate:y = 1.0 * 0.50 * 0.50 - 1.0 * 0.50 - 1.0 or y = -1.250

Parabola, Graphing Vertex & X-Intercepts :

Root plot for : y = x2-x-1 Axis of Symmetry (dashed) x= 0.50 Vertex at x,y = 0.50,-1.25 x-Intercepts (Roots) : Root 1 at x,y = -0.62, 0.00 Root 2 at x,y = 1.62, 0.00

Solve Quadratic Equation by Completing The Square

2.2Solvingx2-x-1 = 0 by Completing The Square.Add 1 lớn both side of the equation : x2-x = 1Now the clever bit: Take the coefficient of x, which is 1, divide by two, giving một nửa, & finally square it giving 1/4Add 1/4 lớn both sides of the equation :On the right h& side we have:1+1/4or, (1/1)+(1/4)The comtháng denominator of the two fractions is 4Adding (4/4)+(1/4) gives 5/4So adding lớn both sides we finally get:x2-x+(1/4) = 5/4Adding 1/4 has completed the left h& side into a perfect square :x2-x+(1/4)=(x-(1/2))•(x-(1/2))=(x-(1/2))2 Things which are equal khổng lồ the same thing are also equal to one another. Sincex2-x+(1/4) = 5/4 andx2-x+(1/4) = (x-(1/2))2 then, according to the law of transitivity,(x-(1/2))2 = 5/4We"ll refer lớn this Equation as Eq. #2.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.Note that 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 Root Principle to lớn Eq.#2.2.1 we get:x-(1/2)= √ 5/4 Add một nửa to lớn both sides to lớn obtain:x = 1/2 + √ 5/4 Since a square root has two values, one positive sầu & the other negativex2 - x - 1 = 0has two solutions:x = 1/2 + √ 5/4 orx = 50% - √ 5/4 lưu ý that √ 5/4 can be written as√5 / √4which is √5 / 2

Solve Quadratic Equation using the Quadratic Formula

2.3Solvingx2-x-1 = 0 by the Quadratic Formula.According to the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B & C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 1B= -1C= -1 Accordingly,B2-4AC=1 - (-4) = 5Applying the quadratic formula : 1 ± √ 5 x=————2 √ 5 , rounded to lớn 4 decimal digits, is 2.2361So now we are looking at:x=(1± 2.236 )/2Two real solutions:x =(1+√5)/2= 1.618 or:x =(1-√5)/2=-0.618