Menu Close

In-a-quadratic-equation-ax-2-bx-c-0-a-b-c-are-distinct-primes-and-the-product-of-the-sum-of-the-roots-and-product-of-the-roots-is-91-9-Find-the-absolute-value-of-difference-between-the-sum-o




Question Number 76401 by saupriyadip571@gmail.com last updated on 27/Dec/19
In a quadratic equation ax^2 −bx+c=0,  a,b, c  are distinct primes and the product  of the sum of the roots and product of  the roots is ((91)/9) . Find the absolute value of  difference between the sum of the roots  and the product of the roots.
$$\mathrm{In}\:\mathrm{a}\:\mathrm{quadratic}\:\mathrm{equation}\:{ax}^{\mathrm{2}} −{bx}+{c}=\mathrm{0}, \\ $$$${a},{b},\:{c}\:\:\mathrm{are}\:\mathrm{distinct}\:\mathrm{primes}\:\mathrm{and}\:\mathrm{the}\:\mathrm{product} \\ $$$$\mathrm{of}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{and}\:\mathrm{product}\:\mathrm{of} \\ $$$$\mathrm{the}\:\mathrm{roots}\:\mathrm{is}\:\frac{\mathrm{91}}{\mathrm{9}}\:.\:\mathrm{Find}\:\mathrm{the}\:\mathrm{absolute}\:\mathrm{value}\:\mathrm{of} \\ $$$$\mathrm{difference}\:\mathrm{between}\:\mathrm{the}\:\mathrm{sum}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots} \\ $$$$\mathrm{and}\:\mathrm{the}\:\mathrm{product}\:\mathrm{of}\:\mathrm{the}\:\mathrm{roots}. \\ $$
Answered by mr W last updated on 27/Dec/19
say roots are α, β.  α+β=(b/a)  αβ=(c/a)  given: (α+β)αβ=((91)/9)  ⇒((bc)/a^2 )=((91)/9)=((13×7)/3^2 )  since a,b,c are prime,  ⇒a=3, b=13 (or 7), c=7 (or 13)  X=(α+β)−αβ=((b−c)/a)=((13−7)/3)=2  ⇒answer is 2.
$${say}\:{roots}\:{are}\:\alpha,\:\beta. \\ $$$$\alpha+\beta=\frac{{b}}{{a}} \\ $$$$\alpha\beta=\frac{{c}}{{a}} \\ $$$${given}:\:\left(\alpha+\beta\right)\alpha\beta=\frac{\mathrm{91}}{\mathrm{9}} \\ $$$$\Rightarrow\frac{{bc}}{{a}^{\mathrm{2}} }=\frac{\mathrm{91}}{\mathrm{9}}=\frac{\mathrm{13}×\mathrm{7}}{\mathrm{3}^{\mathrm{2}} } \\ $$$${since}\:{a},{b},{c}\:{are}\:{prime}, \\ $$$$\Rightarrow{a}=\mathrm{3},\:{b}=\mathrm{13}\:\left({or}\:\mathrm{7}\right),\:{c}=\mathrm{7}\:\left({or}\:\mathrm{13}\right) \\ $$$${X}=\left(\alpha+\beta\right)−\alpha\beta=\frac{{b}−{c}}{{a}}=\frac{\mathrm{13}−\mathrm{7}}{\mathrm{3}}=\mathrm{2} \\ $$$$\Rightarrow{answer}\:{is}\:\mathrm{2}. \\ $$

Leave a Reply

Your email address will not be published. Required fields are marked *