Question

one of the roots of a quadratic equations is 4=√7 find the equation

Answer 1

Irrational Conjugate Theorem

If x=a+b√x=a+b is a root of an arbitrary polynomial equation p(x)=0p(x)=0, then x=a−b√x=a−b is also a root of p(x)=0p(x)=0

If αα and ββ are roots of a quadratic equation, then using Vieta’s rules the quadratic equation is given by x2−(α+β)x+αβ=0x2−(α+β)x+αβ=0

α=−4+7–√⟹β=−4−7–√(Irrational Conjugate Theorem)α=−4+7⟹β=−4−7(Irrational Conjugate Theorem)

x2−(α+β)x+αβx2−(−4+7–√−4−7–√)x+(−4+7–√)(−4−7–√)x2+8x−(7–√+4)(7–√−4)x2+8x−(7−16)x2+8x+9=0=0=0=0=0