=(31+√-479)/-16=31/-16-i/16√ 479 = -1.9375+1.3679i
Rearrange:
Rearrange the equation by subtracting what is to the right of the equal sign from both sides of the equation :
9*(x-5)-(8*(x+5)*x)=0
Step by step solution :
Step 1 :
Equation at the end of step 1 :
(9 • (x - 5)) - (8 • (x + 5) • x) = 0
Step 2 :
Equation at the end of step 2 :
(9 • (x - 5)) - 8x • (x + 5) = 0
Step 3 :
Equation at the end of step 3 :
9 • (x - 5) - 8x • (x + 5) = 0
Step 4 :
Step 5 :
Pulling out like terms :
5.1 Pull out like factors :
-8x2 - 31x - 45 = -1 • (8x2 + 31x + 45)
Trying to factor by splitting the middle term
5.2 Factoring 8x2 + 31x + 45
The first term is, 8x2 its coefficient is 8 .
The middle term is, +31x its coefficient is 31 .
The last term, "the constant", is +45
Step-1 : Multiply the coefficient of the first term by the constant 8 • 45 = 360
Step-2 : Find two factors of 360 whose sum equals the coefficient of the middle term, which is 31 .
-360 + -1 = -361
-180 + -2 = -182
-120 + -3 = -123
-90 + -4 = -94
-72 + -5 = -77
-60 + -6 = -66
For tidiness, printing of 42 lines which failed to find two such factors, was suppressed
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 5 :
-8x2 - 31x - 45 = 0
Step 6 :
Parabola, Finding the Vertex :
6.1 Find the Vertex of y = -8x2-31x-45
Parabolas have a highest or a lowest point called the Vertex . Our parabola opens down and accordingly has a highest point (AKA absolute maximum) . We know this even before plotting "y" because the coefficient of the first term, -8 , is negative (smaller 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 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 to be able to 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 -1.9375
Plugging into the parabola formula -1.9375 for x we can calculate the y -coordinate :
y = -8.0 * -1.94 * -1.94 - 31.0 * -1.94 - 45.0
or y = -14.969
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = -8x2-31x-45
Axis of Symmetry (dashed) {x}={-1.94}
Vertex at {x,y} = {-1.94,-14.97}
Function has no real roots
Solve Quadratic Equation by Completing The Square
6.2 Solving -8x2-31x-45 = 0 by Completing The Square .
Multiply both sides of the equation by (-1) to obtain positive coefficient for the first term:
8x2+31x+45 = 0 Divide both sides of the equation by 8 to have 1 as the coefficient of the first term :
x2+(31/8)x+(45/8) = 0
Subtract 45/8 from both side of the equation :
x2+(31/8)x = -45/8
Now the clever bit: Take the coefficient of x , which is 31/8 , divide by two, giving 31/16 , and finally square it giving 961/256
Add 961/256 to both sides of the equation :
On the right hand side we have :
-45/8 + 961/256 The common denominator of the two fractions is 256 Adding (-1440/256)+(961/256) gives -479/256
So adding to both sides we finally get :
x2+(31/8)x+(961/256) = -479/256
Adding 961/256 has completed the left hand side into a perfect square :
x2+(31/8)x+(961/256) =
(x+(31/16)) • (x+(31/16)) =
(x+(31/16))2
Things which are equal to the same thing are also equal to one another. Since
x2+(31/8)x+(961/256) = -479/256 and
x2+(31/8)x+(961/256) = (x+(31/16))2
then, according to the law of transitivity,
(x+(31/16))2 = -479/256
We'll refer to this Equation as Eq. #6.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x+(31/16))2 is
(x+(31/16))2/2 =
(x+(31/16))1 =
x+(31/16)
Now, applying the Square Root Principle to Eq. #6.2.1 we get:
x+(31/16) = √ -479/256
Subtract 31/16 from both sides to obtain:
x = -31/16 + √ -479/256
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Since a square root has two values, one positive and the other negative
x2 + (31/8)x + (45/8) = 0
has two solutions:
x = -31/16 + √ 479/256 • i
or
x = -31/16 - √ 479/256 • i
Note that √ 479/256 can be written as
√ 479 / √ 256 which is √ 479 / 16
Solve Quadra
Step-by-step explanation:
