Write whether the following statements are true or false. Justify.
(i) Every quadratic has exactly one root.
(ii) Every quadratic has at least one real root.
(iii) Every quadratic has at least two roots.
(iv) Every quadratic has at most two roots.
(v) If coefficient of \(x^2\) and constant term have opposite signs, the quadratic has real roots.
(vi) If coefficient of \(x^2\) and constant have same sign and coefficient of x term is 0, then quadratic has no real roots.
(i) False
(ii) False
(iii) True (but not necessarily distinct real)
(iv) True
(v) True
(vi) True
(i) A quadratic equation can have 0, 1, or 2 real roots. Example: \(x^2+1=0\) has 0 real roots, \((x-2)^2=0\) has 1 real root, and \(x^2-1=0\) has 2 real roots. So, it is not true that every quadratic has exactly one root.
(ii) Some quadratics do not have any real root. Example: \(x^2+1=0\). Here, no real number squared gives \(-1\). So, not every quadratic has at least one real root. This is false.
(iii) By the Fundamental Theorem of Algebra, a quadratic always has 2 roots (they may be real or complex). Example: \(x^2+1=0\) has two roots: \(i\) and \(-i\), which are complex. So the statement “every quadratic has at least two roots” is true if we count complex roots as well.
(iv) Since the highest power of \(x\) is 2, a quadratic can never have more than 2 roots. So, “at most 2 roots” is true.
(v) If the coefficient of \(x^2\) (say \(a\)) and the constant term (say \(c\)) have opposite signs, then their product \(a \times c < 0\). This means the parabola will cut the x-axis at two points, giving two real roots. Example: \(x^2 - 4 = 0\). Here \(a=1\), \(c=-4\), opposite signs → real roots \(x=\pm 2\). So this is true.
(vi) If \(a\) and \(c\) have the same sign, and coefficient of \(x\) (that is \(b\)) is 0, then the quadratic looks like \(ax^2 + c = 0\). Example: \(x^2 + 4 = 0\). Here both \(a\) and \(c\) are positive. This has no real solution because \(x^2 = -4\) is impossible for real numbers. So this statement is true.