The difference between the roots of the quadratic equation x^2−20x+q=0 is 6. find q.

q = 91

Step-by-step explanation:

We are going to use formula for finding roots to solve this questions;

Using this two formula;

ax²  +  bx  + c = 0(1)

Let  ∝  and  β be the two roots of the quadratic equation above, then

x²   -  (∝ + β)x   + ∝β  =  0  (2)

comparing the two equations above

sum of the roots of a quadratic equation is : ∝ + β = -b/a (3)

Products of the root of a quadratic equation is : ∝β = c/a (4)

We can now start solving our question;

The difference between the roots of the quadratic equation x^2−20x+q=0 is 6. Find q.

Let the two roots be  ∝  and  ∝ - 6

Sum of the roots =  ∝ +  ∝ - 6 = 2∝ - 6 (5)

Product of the roots =  ∝( ∝ -  6)  =  ∝²  -   6∝ (6)

From equation (3) and equation (4)

sum of a quadratic equation = -b/a

Product of a quadratic equation = c/a

so;

From equation (5) and (6)

Sum of the roots = 2∝ - 6 = -b/a  (7)

Product of the roots =   ∝²  -   6∝ = c/a  (8)

But again the equation given to us is;

x²−20x+q=0

comparing this with the standard equation, equation(1)

   a = 1           b = 20    and    c = q

From equation (7)  :  2∝ - 6  = -b/a

                                  2∝ - 6 = -20/1

                                   2∝ - 6 = -20

                                   Add  6 to both-side of the equation

                                    2∝ - 6 + 6 = -20+ 6

                                     2∝= -14

                                     Divide both-side of the equation by 2

                                      ∝ =  -7

Also, from equation (8) :   ∝²  -   6∝ = c/a

                                            ∝²  -   6∝ = q/1

                                              ∝²  -   6∝ = q

But  ∝  = -7, so we will substitute  ∝ = -7 in the above equation to get the value of q;

∝²  -   6∝ = q

(-7)²  - 6(-7) = q

49 + 42 = q

91  =  q

Therefore the value of q = 91

[To test the correctness of our answer, lets substitute q back into the equation and solve;

x²−20x+91=0

x² - 7x - 13x + 91 = 0

x(x-7) -13(x-7) = 0

(x-7)(x-13)= 0

x = 7 or x=13

The difference between 7 and 13 is 6.]

Therefore the value of q is 91

y=mx+b is the final answer!   : )

5104.38 inches

step-by-step explanation: