Reformatting the đầu vào :

Changes made to lớn your đầu vào 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 x2-12x-3 The first term is, x2 its coefficient is 1.The middle term is, -12x its coefficient is -12.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 -12.

-3+1=-2
-1+3=2

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

Equation at the end of step 1 :

x2 - 12x - 3 = 0

Step 2 :

Parabola, Finding the Vertex:2.1Find the Vertex ofy = x2-12x-3Parabolas 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 (greater than zero).Each parabola has a vertical line of symmetry that passes through its vertex. 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. For this reason we want to 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 6.0000Plugging into the parabola formula 6.0000 for x we can calculate the y-coordinate:y = 1.0 * 6.00 * 6.00 - 12.0 * 6.00 - 3.0 or y = -39.000

Parabola, Graphing Vertex & X-Intercepts :

Root plot for : y = x2-12x-3 Axis of Symmetry (dashed) x= 6.00 Vertex at x,y = 6.00,-39.00 x-Intercepts (Roots) : Root 1 at x,y = -0.24, 0.00 Root 2 at x,y = 12.24, 0.00

Solve Quadratic Equation by Completing The Square

2.2Solvingx2-12x-3 = 0 by Completing The Square.Add 3 to both side of the equation : x2-12x = 3Now the clever bit: Take the coefficient of x, which is 12, divide by two, giving 6, & finally square it giving 36Add 36 to both sides of the equation :On the right hand side we have:3+36or, (3/1)+(36/1)The common denominator of the two fractions is 1Adding (3/1)+(36/1) gives 39/1So adding lớn both sides we finally get:x2-12x+36 = 39Adding 36 has completed the left hand side into a perfect square :x2-12x+36=(x-6)•(x-6)=(x-6)2 Things which are equal to the same thing are also equal to one another. Sincex2-12x+36 = 39 andx2-12x+36 = (x-6)2 then, according to the law of transitivity,(x-6)2 = 39We"ll refer khổng lồ this Equation as Eq.


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#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-6)2 is(x-6)2/2=(x-6)1=x-6Now, applying the Square Root Principle khổng lồ Eq.#2.2.1 we get:x-6= √ 39 địa chỉ 6 lớn both sides to obtain:x = 6 + √ 39 Since a square root has two values, one positive & the other negativex2 - 12x - 3 = 0has two solutions:x = 6 + √ 39 orx = 6 - √ 39

Solve Quadratic Equation using the Quadratic Formula

2.3Solvingx2-12x-3 = 0 by the Quadratic Formula.According khổng lồ 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=-12C= -3 Accordingly,B2-4AC=144 - (-12) = 156Applying the quadratic formula : 12 ± √ 156 x=——————2Can √ 156 be simplified ?Yes!The prime factorization of 156is2•2•3•13 to be able lớn remove something from under the radical, there have to be 2 instances of it (because we are taking a square i.e. Second root).√ 156 =√2•2•3•13 =±2 •√ 39 √ 39 , rounded lớn 4 decimal digits, is 6.2450So now we are looking at:x=(12±2• 6.245 )/2Two real solutions:x =(12+√156)/2=6+√ 39 = 12.245 or:x =(12-√156)/2=6-√ 39 = -0.245

Two solutions were found :

x =(12-√156)/2=6-√ 39 = -0.245 x =(12+√156)/2=6+√ 39 = 12.245