Question

If a+√b is one of the roots of a quadratic equation, then the other root is​

Answer 1

Question:-

If a+√b is one of the roots of a quadratic equation, then the other root is

Solution:-

Suppose the given quadratic equation is px² + qx + r= 0, where a , b and c are rational numbers.

If α and β are two roots of the above equation with α = a+√b.

Now α + β = -q/p, which is rational.

Also, αβ = r/p, which is also rational.

Since, α = a+√b is irrational and sum as well as product of α and β is rational, so β must be an irrational number and that irrational number must be a-√b, so that

α + β = (a+√b) + (a-√b) = 2a, a rational number and, αβ = (a+√b)(a-√b) = a² - b, a rational number.

∴ So, the other root of a quadratic equation having the one root as (a+√b) is (a-√b), where a and b are rational numbers.

Answer 2

Answer:

Suppose the given quadratic equation is px² + qx + r= 0, where a , b and c are rational numbers.

If α and β are two roots of the above equation with α = a+√b.

Now α + β = -q/p, which is rational.

Also, αβ = r/p, which is also rational.

Since, α = a+√b is irrational and sum as well as product of α and β is rational, so β must be an irrational number and that irrational number must be a-√b, so that

α + β = (a+√b) + (a-√b) = 2a, a rational number and, αβ = (a+√b)(a-√b) = a² - b, a rational number.

So, the other root of a quadratic equation having the one root as (a+√b) is (a-√b), where a and b are rational numbers.