Write whether the following statements are true or false. Justify your answers.
(i) Every quadratic equation has exactly one root.
(ii) Every quadratic equation has at least one real root.
(iii) Every quadratic equation has at least two roots.
(iv) Every quadratic equation has at most two roots.
(v) If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots.
(vi) If the coefficient of x2 and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots.
(i) Every quadratic equation has exactly one root: False
A quadratic equation has exactly two roots.
(ii) Every quadratic equation has at least one real root: False
A quadratic equation may have real or imaginary roots.
(iii) Every quadratic equation has at least two roots: False
A quadratic equation has exactly two roots, no more no less.
(iv) Every quadratic equation has at most two roots: True
A quadratic equation can’t have more than two roots.
(v) If the coefficient of x2 and the constant term of a quadratic equation have opposite signs, then the quadratic equation has real roots: True
The standard form of a quadratic equation is given by,
ax2 + bx + c = 0, a ≠ 0
So, if coefficient of x2(a) and constant term(c) have opposite signs then, the roots will always be real i.e.,
ac < 0
⇒ b2 - 4ac > 0
(vi) If the coefficient of x2 and the constant term have the same sign and if the coefficient of x term is zero, then the quadratic equation has no real roots: True
The standard form of a quadratic equation is given by,
ax2 + bx + c = 0, a ≠ 0
So, if coefficient of x2(a) and constant term(c) have the same sign and coefficient of x(b) is zero, then the roots will be imaginary.
ac > 0
⇒ b2 - 4ac < 0