Question

If sum of roots is 2 and product is 5, then the quadratic equation is
( a ) ${{x}^{2}}+2x+5$
( b ) ${{x}^{2}}-2x-5$
( c ) ${{x}^{2}}+2x-5$
( d ) ${{x}^{2}}-2x+5$

Answer 1

Hint: There are various methods to do this question by short cut method or by first finding roots of quadratic equations and then their sum and product. The concept we will use here is a short cut method very well known as finding sum and product of roots using coefficients of quadratic equations.

Complete step by step answer:
What we do in this method is if we have a quadratic equation \[a{{x}^{2}}+bx+c=0\] and let $\alpha $ and $\beta $ be roots of this quadratic polynomial. Then we can easily find the sum of roots and product of roots without even finding the roots of polynomials as, $\alpha +\beta $ which represents sum of roots is equals to $-\dfrac{b}{a}$ , where b is linear quadratic coefficient and a is quadratic coefficient of quadratic equation \[a{{x}^{2}}+bx+c=0\]. And $\alpha \cdot \beta $ which represents product of roots is equals to $\dfrac{c}{a}$ , where c is constant coefficient and a is quadratic coefficient of quadratic equation \[a{{x}^{2}}+bx+c=0\].
Now, for all quadratic polynomials given in option, we can find their sum of roots and product of roots accordingly as discussed above. Given that sum of roots is 2 and product of roots is 5
For quadratic equation, ${{x}^{2}}+2x+5$, sum of root will be given as $-\dfrac{2}{1}=-2$, where 2 is linear coefficient and 1 is quadratic coefficient of quadratic equation and product of root will be given as $\dfrac{5}{1}=5$, where 5 is constant coefficient and 1 is quadratic coefficient of quadratic equation, which is not equal to given sum and product of roots.
For quadratic equation, ${{x}^{2}}-2x-5$, sum of root will be given as $-\dfrac{(-2)}{1}=2$, where -2 is linear coefficient and 1 is quadratic coefficient of quadratic equation and product of root will be given as $\dfrac{-5}{1}=-5$ which is not equal to given sum and product of roots.
For quadratic equation, ${{x}^{2}}+2x-5$, sum of root will be given as $-\dfrac{2}{1}=-2$, where 2 is linear coefficient and 1 is quadratic coefficient of quadratic equation and product of root will be given as $\dfrac{-5}{1}=-5$ which is not equal to given sum and product of roots.
For quadratic equation, ${{x}^{2}}-2x+5$, sum of root will be given as $-\dfrac{(-2)}{1}=2$, where -2 is linear coefficient and 1 is quadratic coefficient of quadratic equation and product of root will be given as $\dfrac{5}{1}=5$ which is equal to given sum and product of roots.

So, the correct answer is “Option d”.

Note: Do not forget to write the statement in solutions. Be careful about signs in the formula which you are using to solve the question. You can also solve it by first finding the roots of the equation and then sum and product of roots and eliminate the wrong options.