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

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

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

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

If we multiply equation 2 by 3^z and add 1 in the smallest root of the equation newly formed, then find the resultant number?

The correct answer is 2.

Let us solve the problem step-by-step:

Step 1: Understand the roots of the given equations using Vieta's formulas

We are given Equation 1:
$a^2-7a+d=0$
The roots of Equation 1 are given as x and y. Using the relationship between roots and coefficients (Vieta's formulas):
Sum of roots: $x+y=7$ (Equation A)
Product of roots: $x\cdot y=d$ (Equation B)

We are also given Equation 2:
$b^2���4b+(d-9)=0$
The roots of Equation 2 are given as x and (y - x). Using Vieta's formulas:
Sum of roots: $x+(y-x)=y=4$ (Equation C)
Product of roots: $x\cdot (y-x)=d-9$ (Equation D)

Step 2: Find the values of x, y, and the roots of Equation 2

From Equation C, we have:
$y=4$

Substitute $y=4$ into Equation A:
$x+4=7\Rightarrow x=3$

Now, we can find the roots of Equation 2:
First root = $x=3$
Second root = $y-x=4-3=1$
The roots of Equation 2 are 3 and 1. The smallest root of Equation 2 is 1.

Step 3: Analyze the effect of multiplying the equation by 3^z

When we multiply a quadratic equation of the form $f\left(b\right)=0$ by a non-zero constant $3^z$ , the newly formed equation is:
$3^z\cdot [b^2-4b+(d-9\left)\right]=0$
Since $3^z\ne 0$ for all real values of z, the roots of the equation remain completely unchanged.
Therefore, the roots of the newly formed equation are still 1 and 3, and the smallest root remains 1.

Step 4: Find the final resultant number

Adding 1 to the smallest root of the newly formed equation:
$1+1=2$

Thus, the resultant number is 2.