Question

The numerical difference between the roots of '${\mathrm{x}}^2-7\mathrm{x}-9=0$' is:

• A. $-2$
• B. $16$
• C. $\sqrt{85}$
• D. $2\sqrt{85}$

Answer 1

The correct option is C

$\sqrt{85}$

Given: ${\mathrm{x}}^2-7\mathrm{x}-9=0$

Let '$\mathrm{a}$‘ and ’$\mathrm{b}$' be the roots.

Step 1: Find the sum of roots:

$\Rightarrow$The formula for the sum of roots of quadratic equation$=\frac{-\mathrm{coefficient}\mathrm{of}\mathrm{x}}{\mathrm{coefficient}\mathrm{of}{\mathrm{x}}^2}$

$\therefore \mathrm{a}+\mathrm{b}=\frac{-\left(-7\right)}{1}$

$=7$

Step 2: Find the product of roots:

$\Rightarrow$The formula for the product of roots of quadratic$=\frac{\mathrm{constant}\mathrm{term}}{\mathrm{coefficient}\mathrm{of}{\mathrm{x}}^2}$

$\therefore \mathrm{a}\times \mathrm{b}=\frac{-9}{1}$

$=-9$

Step 3: Use the identity,'${\left(\mathrm{a}-\mathrm{b}\right)}^2={\left(\mathrm{a}+\mathrm{b}\right)}^2-4\mathrm{ab}$, find the value of $\mathrm{a}-\mathrm{b}$

Putting the values of the sum of roots and the product of roots in the above equation:

$\begin{array}{rcl}\therefore {\left(\mathrm{a}-\mathrm{b}\right)}^2& =& {\left(7\right)}^2-4\times \left(-9\right)\\ & =& 49+36\\ & =& 85\\ & & \end{array}$

$\therefore \mathrm{a}-\mathrm{b}=\pm \sqrt{85}$

But we don't have $-\sqrt{85}$ in options,

So, $\mathrm{a}-\mathrm{b}=\sqrt{85}$

Therefore, the correct option is C.