p² - 5p-1=0 find (P-1/p)
Answers
Step-by-step explanation:
Reformatting the input :
Changes made to your input should not affect the solution:
(1): "p2" was replaced by "p^2".
Step by step solution :
STEP
1
:
STEP
2
:
Pulling out like terms
2.1 Pull out like factors :
-p2 - 5p + 1 = -1 • (p2 + 5p - 1)
Trying to factor by splitting the middle term
2.2 Factoring p2 + 5p - 1
The first term is, p2 its coefficient is 1 .
The middle term is, +5p its coefficient is 5 .
The last term, "the constant", is -1
Step-1 : Multiply the coefficient of the first term by the constant 1 • -1 = -1
Step-2 : Find two factors of -1 whose sum equals the coefficient of the middle term, which is 5 .
-1 + 1 = 0
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step
2
:
-p2 - 5p + 1 = 0
STEP
3
:
Parabola, Finding the Vertex:
3.1 Find the Vertex of y = -p2-5p+1
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, -1 , 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,Ap2+Bp+C,the p -coordinate of the vertex is given by -B/(2A) . In our case the p coordinate is -2.5000
Plugging into the parabola formula -2.5000 for p we can calculate the y -coordinate :
y = -1.0 * -2.50 * -2.50 - 5.0 * -2.50 + 1.0
or y = 7.250
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = -p2-5p+1
Axis of Symmetry (dashed) {p}={-2.50}
Vertex at {p,y} = {-2.50, 7.25}
p -Intercepts (Roots) :
Root 1 at {p,y} = { 0.19, 0.00}
Root 2 at {p,y} = {-5.19, 0.00}
3.2 Solving -p2-5p+1 = 0 by Completing The Square .
Multiply both sides of the equation by (-1) to obtain positive coefficient for the first term:
p2+5p-1 = 0 Add 1 to both side of the equation :
p2+5p = 1
Now the clever bit: Take the coefficient of p , which is 5 , divide by two, giving 5/2 , and finally square it giving 25/4
Add 25/4 to both sides of the equation :
On the right hand side we have :
1 + 25/4 or, (1/1)+(25/4)
The common denominator of the two fractions is 4 Adding (4/4)+(25/4) gives 29/4
So adding to both sides we finally get :
p2+5p+(25/4) = 29/4
Adding 25/4 has completed the left hand side into a perfect square :
p2+5p+(25/4) =
(p+(5/2)) • (p+(5/2)) =
(p+(5/2))2
Things which are equal to the same thing are also equal to one another. Since
p2+5p+(25/4) = 29/4 and
p2+5p+(25/4) = (p+(5/2))2
then, according to the law of transitivity,
(p+(5/2))2 = 29/4
We'll refer to 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
(p+(5/2))2 is
(p+(5/2))2/2 =
(p+(5/2))1 =
p+(5/2)
Now, applying the Square Root Principle to Eq. #3.2.1 we get:
p+(5/2) = √ 29/4
Subtract 5/2 from both sides to obtain:
p = -5/2 + √ 29/4
Since a square root has two values, one positive and the other negative
p2 + 5p - 1 = 0
has two solutions:
p = -5/2 + √ 29/4
or
p = -5/2 - √ 29/4
Note that √ 29/4 can be written as
√ 29 / √ 4 which is √ 29 / 2
Solve Quadratic Equation using the Quadratic Formula
3.3 Solving -p2-5p+1 = 0 by the Quadratic Formula .
According to the Quadratic Formula, p , the solution for Ap2+Bp+C = 0 , where A, B and C are numbers, often called coefficients, is given by :
- B ± √ B2-4AC
p = ————————
2A
In our case, A = -1
B = -5
C = 1
Accordingly, B2 - 4AC =
25 - (-4) =
29
Applying the quadratic formula :
5 ± √ 29
p = —————
-2
√ 29 , rounded to 4 decimal digits, is 5.3852
So now we are looking at:
p = ( 5 ± 5.385 ) / -2
Two real solutions:
p =(5+√29)/-2=-5.193
or:
p =(5-√29)/-2= 0.193
Two solutions were found :
p =(5-√29)/-2= 0.193
p =(5+√29)/-2=-5.193