Question

find value of k for which x^2 + 4x + k is a perfect square

is it true that quadratic equations will be a perfect square , only if discrimant is zero? ....if so why is that?
( pls respond...would really appreciate it​

Answer 1

$We \: can \: say \: that \: the \: square \: of \: any \: integers\: will \: be \: a \: perfect \: square \: number. \\ For \: example: \: {1}^{2} = 1, \: {2}^{2} = 4 ,\: {3}^{2} = 9, \: {( - 5)}^{2} = 25, \: and \: so \: on. \: here\: 1 ,\: 4, \: 9, \: 25 \: are \: perfect \: square \: number. \\ And \: also \: suppose \: x \: is \: a n\:integer \: then\: any \: square \: numbers\: are \: in \: the \: form \: of \: {(x)}^{2} = x \times x= some \: perfect \: squares \: {(1 + x)}^{2} =(1 + x) \times (1 + x)= some \: perfect \: squares \: or \: {(1 - x)}^{2} =(1 - x) \times (1 - x)= some \: perfect \: squares and \: so \: many. \: which \: are \: quadratic \: equations \: with \: silmilar \: roots . \\ For \: the \: roots \: of \: a \: quadratic \: eqution : x = \frac{ - b± \sqrt{D} }{a} \: ,where \: D\: is \: discriminant. \\ We \: know \: that \: for \: perfect \: square \: number \: roots \: will \: be \: similar \: (previously \: discussed). \\ \sqrt{D} \: should \: be \: 0 \: for \: the \: equal \: roots. \: \\ \\ then \: we \: can \: use \: previously \: discussed \: things \: so \: \sqrt{D} = 0. \\ \implies{ \sqrt{ {b}^{2} - 4ac } = 0. \: (where \: b \: is \: coefficient \: of \: x \: and\: a \: is \: coefficient \: of \: {x}^{2}). } \\ so \: substituting \: values: \sqrt{ {4}^{2} - 4 \times1 \times k } = 0. \\ \implies{16 - 4k = 0} \\ \implies{k = \frac{16}{4} } \\ \implies{k \: = \: 4.}$