Reformatting the input :

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

Step by step solution :

Step 1 :

Trying khổng lồ factor by splitting the middle term

1.1Factoring x2-4x-3 The first term is, x2 its coefficient is 1.The middle term is, -4x its coefficient is -4.The last term, "the constant", is -3Step-1 : Multiply the coefficient of the first term by the constant 1•-3=-3Step-2 : Find two factors of -3 whose sum equals the coefficient of the middle term, which is -4.


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

Equation at the over of step 1 :

x2 - 4x - 3 = 0

Step 2 :

Parabola, Finding the Vertex:2.1Find the Vertex ofy = x2-4x-3Parabolas have sầu a highest or a lowest point called the Vertex.Our parabola opens up and 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 (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 sầu 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 khổng lồ be able to lớn 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 2.0000Plugging inlớn the parabola formula 2.0000 for x we can calculate the y-coordinate:y = 1.0 * 2.00 * 2.00 - 4.0 * 2.00 - 3.0 or y = -7.000

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = x2-4x-3 Axis of Symmetry (dashed) x= 2.00 Vertex at x,y = 2.00,-7.00 x-Intercepts (Roots) : Root 1 at x,y = -0.65, 0.00 Root 2 at x,y = 4.65, 0.00

Solve Quadratic Equation by Completing The Square

2.2Solvingx2-4x-3 = 0 by Completing The Square.Add 3 lớn both side of the equation : x2-4x = 3Now the clever bit: Take the coefficient of x, which is 4, divide by two, giving 2, & finally square it giving 4Add 4 lớn both sides of the equation :On the right hvà side we have:3+4or, (3/1)+(4/1)The comtháng denominator of the two fractions is 1Adding (3/1)+(4/1) gives 7/1So adding to both sides we finally get:x2-4x+4 = 7Adding 4 has completed the left hand side into lớn a perfect square :x2-4x+4=(x-2)•(x-2)=(x-2)2 Things which are equal lớn the same thing are also equal to lớn one another. Sincex2-4x+4 = 7 andx2-4x+4 = (x-2)2 then, according to the law of transitivity,(x-2)2 = 7We"ll refer to lớn this Equation as Eq. #2.2.1 The Square Root Principle says that When two things are equal, their square roots are equal.cảnh báo that the square root of(x-2)2 is(x-2)2/2=(x-2)1=x-2Now, applying the Square Root Principle to lớn Eq.#2.2.1 we get:x-2= √ 7 Add 2 to both sides to lớn obtain:x = 2 + √ 7 Since a square root has two values, one positive sầu & the other negativex2 - 4x - 3 = 0has two solutions:x = 2 + √ 7 orx = 2 - √ 7

Solve sầu Quadratic Equation using the Quadratic Formula

2.3Solvingx2-4x-3 = 0 by the Quadratic Formula.According khổng lồ the Quadratic Formula,x, the solution forAx2+Bx+C= 0 , where A, B and C are numbers, often called coefficients, is given by :-B± √B2-4ACx = ————————2A In our case,A= 1B= -4C= -3 Accordingly,B2-4AC=16 - (-12) = 28Applying the quadratic formula : 4 ± √ 28 x=—————2Can √ 28 be simplified ?Yes!The prime factorization of 28is2•2•7 To be able to remove something from under the radical, there have lớn be 2 instances of it (because we are taking a square i.e. second root).√ 28 =√2•2•7 =±2 •√ 7 √ 7 , rounded to 4 decimal digits, is 2.6458So now we are looking at:x=(4±2• 2.646 )/2Two real solutions:x =(4+√28)/2=2+√ 7 = 4.646 or:x =(4-√28)/2=2-√ 7 = -0.646

Two solutions were found :

x =(4-√28)/2=2-√ 7 = -0.646 x =(4+√28)/2=2+√ 7 = 4.646

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