Question

If one of the roots of the quadratic equation is $p + \sqrt q $ then another root will be $p - \sqrt q $. This statement is true if the coefficients are rational.

Answer 1

Hint: In order to prove that $p - \sqrt q $ will be another root of the quadratic equation we need to consider the general form of quadratic equation, that is, $ax_{}^2 + bx + c$$ = 0$.
Let us assume that $p + \sqrt q $ is a surd root of $ax_{}^2 + bx + c$$ = 0$, so we can say that the value of $ax_{}^2 + bx + c$$ = 0$ must be fulfilled by $x = p + \sqrt q $

Complete step-by-step answer:
So considering $p + \sqrt q $ as one of the surd root of $ax_{}^2 + bx + c$$ = 0$ and substituting the value of $x = p + \sqrt q $ in $ax_{}^2 + bx + c$$ = 0$
We can write it as, $a(p + \sqrt {q)} _{}^2 + b(p + \sqrt q ) + c = 0$
Now we have to apply the formula of $(a + b)_{}^2 = a_{}^2 + b_{}^2 + 2ab$ in $(p + \sqrt {q)} _{}^2$ to simplify the sum and we get,
$a$$(p_{}^2 + q + 2p\sqrt q ) + bp + b\sqrt q + c$$ = 0$
On multiply $a$ with $p_{}^2 + q + 2p\sqrt q $,
$ap_{}^2 + aq + 2ap\sqrt q + bp + b\sqrt q + c = 0$
Now in this step we will take $\sqrt q $ as common,
We get, $ap_{}^2 + aq + bp + c + (2ap + b)\sqrt q = 0$
Again substituting the value of $x = p - \sqrt q $ in equation $ax_{}^2 + bx + c$$ = 0$ we get-
$a(p - \sqrt {q)} _{}^2 + b(p - q) + c = 0$
Now by applying the formula of $(a - b)_{}^2 = a_{}^2 + b_{}^2 - 2ab$ we get-
$a$$(p_{}^2 + q - 2p\sqrt q ) + bp - b\sqrt q + c$$ = 0$
Again we have to multiply $a$ with $p_{}^2 + q - 2p\sqrt q $
$ap_{}^2 + aq - 2ap\sqrt q + bp - b\sqrt q + c = 0$
Here, we have to take $q$ as common term for simplifying the equation
$ap_{}^2 + aq + bp + c - (2ap + b)\sqrt q = 0$
Now from the above equations it is clearly shown that $p - \sqrt q $ is another root of $ax_{}^2 + bx + c$$ = 0$ as the equation $ax_{}^2 + bx + c$ $ = 0$ is fulfilled by $x = p - \sqrt q $ when $p + \sqrt q $ is a root of the quadratic equation when the coefficients are rational.

Note: Generally, the quadratic equation is in the form of $ax_{}^2 + bx + c$$ = 0$
In cases where two zeros of the quadratic equations are irrational then the two roots of the quadratic equation will occur in pair which means if one root is $p + \sqrt q $ then the other root will be $p - \sqrt q $
There are two zeros in a quadratic equation which can be found by applying a quadratic formula, factoring or completing a square.