Read the following quadric equation carefully and answer the questions given below.

Equation 1:${\mathrm{ a}}^2-7a+d=0$

Equation 2:${\mathrm{ b}}^2-4b+(d-9)=0$

Note: Roots of the equation 1 are ‘x’ and ‘y’ and the roots of equation 2 are ‘x’ and ‘y – x’ .

Find the value of (d-8)^y

The correct answer is 256.

Let's solve the equations step-by-step to find the value of $(d-8{)}^y$.

First, we are given two quadratic equations:

Equation 1:
$a^2-7a+d=0$

The roots of Equation 1 are given as x and y. Using the relations between the roots and coefficients of a quadratic equation (sum of roots and product of roots):
Sum of roots:
$x+y=7$ --- (Equation 3)
Product of roots:
$xy=d$ --- (Equation 4)

Equation 2:
$b^2-4b+(d-9)=0$

The roots of Equation 2 are given as x and y - x. Using the relations between the roots and coefficients for Equation 2:
Sum of roots:
$x+(y-x)=4$
Simplifying this, we get:
$y=4$ --- (Equation 5)

Now, let's find the value of x by substituting $y=4$ into Equation 3:
$x+4=7$
$x=3$

With both x and y known, we can find the value of d using Equation 4:
$d=xy=3\times 4=12$

Finally, we calculate the required expression $(d-8{)}^y$:
$(d-8{)}^y=(12-8{)}^4$
$=4^4$
$=256$