Question

If a and b are the roots of the quadratic equation ${{x}^{2}}+px+12=0$ with the condition $a-b=1$, then the value of ‘p’ is:
(). 1
(). 7
(). -7
(). +7 or -7

Answer 1

Hint: We will use the formula of sum of roots and product of roots. And with the help of that we can find the value of a + b and ab. Then we will use the formula ${{\left( a-b \right)}^{2}}={{\left( a+b \right)}^{2}}-4ab$ and then we will substitute all the values that we know to find the value of p.

Complete step-by-step answer:

If the equation is $a{{x}^{2}}+bx+c=0$ and roots are $\alpha$ and $\beta$,
Then the formula for sum of roots is: $\alpha +\beta =\dfrac{-b}{a}$
The formula for product of roots is: $\alpha \beta =\dfrac{c}{a}$
Using the above formula for the equation ${{x}^{2}}+px+12=0$ we get,
The sum of roots as: $a+b=-p........(1)$
The product of roots as: $ab=12.........(2)$
Now we have been given that a – b = 1
Now using the formula ${{\left( a-b \right)}^{2}}={{\left( a+b \right)}^{2}}-4ab$ and substituting the values from (1) and (2) we get,
$\begin{aligned} & {{\left( 1 \right)}^{2}}={{\left( -p \right)}^{2}}-4\left( 12 \right) \\\ & 1={{p}^{2}}-48 \\\ & {{p}^{2}}=49 \\\ & p=\pm 7 \\\ \end{aligned}$
Hence, from this we get the value of p as +7 and -7.
Hence, the correct answer is option (d).

Note: The formula for sum of roots and product of roots must be kept in mind. And how we have used ${{\left( a-b \right)}^{2}}={{\left( a+b \right)}^{2}}-4ab$ this formula to solve this question easily. One can also solve this question by finding the value of a and b in terms of p and then substituting it in a – b = 1, and from there also we can find the value of p.