Respuesta :
Answer:
x = 1
x =(7-√77)/2=-0.887
x =(7+√77)/2= 7.887
(((x3) - 23x2) + 17) - 10 = 0
Step 2 :
Polynomial Roots Calculator :
2.1 Find roots (zeroes) of : F(x) = x3-8x2+7
Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0
Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers
The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient
In this case, the Leading Coefficient is 1 and the Trailing Constant is 7.
The factor(s) are:
of the Leading Coefficient : 1
of the Trailing Constant : 1 ,7
Let us test ....
P Q P/Q F(P/Q) Divisor
-1 1 -1.00 -2.00
-7 1 -7.00 -728.00
1 1 1.00 0.00 x-1
7 1 7.00 -42.00
The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms
In our case this means that
x3-8x2+7
can be divided with x-1
Polynomial Long Division :
2.2 Polynomial Long Division
Dividing : x3-8x2+7
("Dividend")
By : x-1 ("Divisor")
dividend x3 - 8x2 + 7
- divisor * x2 x3 - x2
remainder - 7x2 + 7
- divisor * -7x1 - 7x2 + 7x
remainder - 7x + 7
- divisor * -7x0 - 7x + 7
remainder 0
Quotient : x2-7x-7 Remainder: 0
Trying to factor by splitting the middle term
2.3 Factoring x2-7x-7
The first term is, x2 its coefficient is 1 .
The middle term is, -7x its coefficient is -7 .
The last term, "the constant", is -7
Step-1 : Multiply the coefficient of the first term by the constant 1 • -7 = -7
Step-2 : Find two factors of -7 whose sum equals the coefficient of the middle term, which is -7 .
-7 + 1 = -6
-1 + 7 = 6
Observation : No two such factors can be found !!
Conclusion : Trinomial can not be factored
Equation at the end of step 2 :
(x2 - 7x - 7) • (x - 1) = 0
Step 3 :
Theory - Roots of a product :
3.1 A product of several terms equals zero.
When a product of two or more terms equals zero, then at least one of the terms must be zero.
We shall now solve each term = 0 separately
In other words, we are going to solve as many equations as there are terms in the product
Any solution of term = 0 solves product = 0 as well.
Parabola, Finding the Vertex :
3.2 Find the Vertex of y = x2-7x-7
For any parabola,Ax2+Bx+C,the x -coordinate of the vertex is given by -B/(2A) . In our case the x coordinate is 3.5000
Plugging into the parabola formula 3.5000 for x we can calculate the y -coordinate :
y = 1.0 * 3.50 * 3.50 - 7.0 * 3.50 - 7.0
or y = -19.250
Parabola, Graphing Vertex and X-Intercepts :
Root plot for : y = x2-7x-7
Axis of Symmetry (dashed) {x}={ 3.50}
Vertex at {x,y} = { 3.50,-19.25}
x -Intercepts (Roots) :
Root 1 at {x,y} = {-0.89, 0.00}
Root 2 at {x,y} = { 7.89, 0.00}
Solve Quadratic Equation by Completing The Square
3.3 Solving x2-7x-7 = 0 by Completing The Square .
Add 7 to both side of the equation :
x2-7x = 7
Now the clever bit: Take the coefficient of x , which is 7 , divide by two, giving 7/2 , and finally square it giving 49/4
Add 49/4 to both sides of the equation :
On the right hand side we have :
7 + 49/4 or, (7/1)+(49/4)
The common denominator of the two fractions is 4 Adding (28/4)+(49/4) gives 77/4
So adding to both sides we finally get :
x2-7x+(49/4) = 77/4
Adding 49/4 has completed the left hand side into a perfect square :
x2-7x+(49/4) =
(x-(7/2)) • (x-(7/2)) =
(x-(7/2))2
Things which are equal to the same thing are also equal to one another. Since
x2-7x+(49/4) = 77/4 and
x2-7x+(49/4) = (x-(7/2))2
then, according to the law of transitivity,
(x-(7/2))2 = 77/4
We'll refer to this Equation as Eq. #3.3.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(7/2))2 is
(x-(7/2))2/2 =
(x-(7/2))1 =
x-(7/2)
Now, applying the Square Root Principle to Eq. #3.3.1 we get:
x-(7/2) = √ 77/4
Add 7/2 to both sides to obtain:
x = 7/2 + √ 77/4
Since a square root has two values, one positive and the other negative
x2 - 7x - 7 = 0
has two solutions:
x = 7/2 + √ 77/4
or
x = 7/2 - √ 77/4
Note that √ 77/4 can be written as
√ 77 / √ 4 which is √ 77 / 2
Solve Quadratic Equation using the Quadratic Formula
3.4 Solving x2-7x-7 = 0 by the Quadratic Formula .
According to 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 = 1
B = -7
C = -7
Accordingly, B2 - 4AC =
49 - (-28) =
77
Applying the quadratic formula :
7 ± √ 77
x = —————
2
√ 77 , rounded to 4 decimal digits, is 8.7750
So now we are looking at:
x = ( 7 ± 8.775 ) / 2
Two real solutions:
x =(7+√77)/2= 7.887
or:
x =(7-√77)/2=-0.887
Solving a Single Variable Equation :
3.5 Solve : x-1 = 0
Add 1 to both sides of the equation :
x = 1
x = 1
x =(7-√77)/2=-0.887
x =(7+√77)/2= 7.887
