Solve Quadratic equations 4x^2-5x-12=0 Tiger Algebra Solver

Step by step solution :

Step 1 :

Equation at the kết thúc of step 1 :

(2²x² - 5x) - 12 = 0

Step 2 :

Trying lớn factor by splitting the middle term

2.1 Factoring 4x²-5x-12

The first term is, 4x² its coefficient is 4 .
The middle term is, -5x its coefficient is -5 .
The last term, "the constant", is -12

Step-1 : Multiply the coefficient of the first term by the constant 4 • -12 = -48

Step-2 : Find two factors of -48 whose sum equals the coefficient of the middle term, which is -5 .

-48	+	1	=	-47
-24	+	2	=	-22
-16	+	3	=	-13
-12	+	4	=	-8
-8	+	6	=	-2
-6	+	8	=	2
-4	+	12	=	8
-3	+	16	=	13
-2	+	24	=	22
-1	+	48	=	47

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

Equation at the kết thúc of step 2 :

4x² - 5x - 12 = 0

Step 3 :

Parabola, Finding the Vertex :

3.1      Find the Vertex of   y = 4x²-5x-12Parabolas have 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, 4 , is positive (greater than thở 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 model 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 lớn be able 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 0.6250  Plugging into the parabola formula 0.6250 for x we can calculate the y -coordinate :
  y = 4.0 * 0.62 * 0.62 - 5.0 * 0.62 - 12.0
or   y = -13.562

Parabola, Graphing Vertex and X-Intercepts :

Root plot for : y = 4x²-5x-12
Axis of Symmetry (dashed) {x}={ 0.62}
Vertex at {x,y} = { 0.62,-13.56}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-1.22, 0.00}
Root 2 at {x,y} = { 2.47, 0.00}

Solve Quadratic Equation by Completing The Square

3.2 Solving 4x²-5x-12 = 0 by Completing The Square .Divide both sides of the equation by 4 to have 1 as the coefficient of the first term :
x²-(5/4)x-3 = 0

Add 3 lớn both side of the equation :
x²-(5/4)x = 3

Now the clever bit: Take the coefficient of x , which is 5/4 , divide by two, giving 5/8 , and finally square it giving 25/64

Add 25/64 lớn both sides of the equation :
  On the right hand side we have :
   3  +  25/64    or, (3/1)+(25/64)
  The common denominator of the two fractions is 64   Adding (192/64)+(25/64) gives 217/64
  So adding lớn both sides we finally get :
   x²-(5/4)x+(25/64) = 217/64

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Adding 25/64 has completed the left hand side into a perfect square :
   x²-(5/4)x+(25/64)  =
   (x-(5/8)) • (x-(5/8))  =
  (x-(5/8))²
Things which are equal lớn the same thing are also equal lớn one another. Since
   x²-(5/4)x+(25/64) = 217/64 and
   x²-(5/4)x+(25/64) = (x-(5/8))²
then, according lớn the law of transitivity,
   (x-(5/8))² = 217/64

We'll refer lớn this Equation as Eq. #3.2.1

The Square Root Principle says that When two things are equal, their square roots are equal.

Note that the square root of
   (x-(5/8))² is
   (x-(5/8))^2/2 =
  (x-(5/8))¹ =
   x-(5/8)

Now, applying the Square Root Principle to Eq. #3.2.1 we get:
x-(5/8) = √ 217/64

Add 5/8 lớn both sides lớn obtain:
x = 5/8 + √ 217/64

Since a square root has two values, one positive and the other negative
   x² - (5/4)x - 3 = 0
   has two solutions:
  x = 5/8 + √ 217/64
   or
  x = 5/8 - √ 217/64

Note that √ 217/64 can be written as
√ 217 / √ 64 which is √ 217 / 8

Solve Quadratic Equation using the Quadratic Formula

3.3     Solving    4x²-5x-12 = 0 by the Quadratic Formula .According lớn the Quadratic Formula,  x  , the solution for   Ax²+Bx+C  = 0 , where A, B and C are numbers, often called coefficients, is given by :

            - B  ± √ B²-4AC
  x =   ————————
                      2A   In our case,  A   =     4
                      B   =    -5
                      C   =  -12 Accordingly,  B²  -  4AC   =
                     25 - (-192) =
                     217Applying the quadratic formula :

               5 ± √ 217
   x  =    —————
                    8 √ 217 , rounded lớn 4 decimal digits, is 14.7309
So now we are looking at:
           x  =  ( 5 ± 14.731 ) / 8

Two real solutions:

x =(5+√217)/8= 2.466

or:

x =(5-√217)/8=-1.216

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