Question

Find the quadratic equation with rational coefficients which has $5-2\sqrt{5}$ as a root.

• A. $x^2-10x-5=0$
• B. $x^2+10x+5=0$
• C. $x^2-10x+5=0$
• D. None of the above

Answer 1

The correct option is C $x^2-10x+5=0$

According to the problem, coefficients of the required quadratic equation are rational and its one root is $5-2\sqrt{5}$.

We know in a quadratic equation with rational coefficients irrational roots occur in conjugate pairs.

Since equation has rational coefficients, the other root is $5+2\sqrt{5}$.

Now, sum of roots $=S=(5-2\sqrt{5})+(5+2\sqrt{5})=10$

and product of roots $=P=(5-2\sqrt{5})(5+2\sqrt{5})=25-(2\sqrt{5}{)}^2=25-20=5$

Hence, required quadratic equation is $x^2-Sx+P=0$ i.e. $x^2-10x+5=0$.