Use the x-intercept method to find all real solutions of the equation.
x^3-8x^2+17-10=0

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

898.1666667

step-by-step explanation:

5389/6 = 898.1666667