Question

For what value of k the quadratic equation x2-4x+k=0 has real roots

Answer 1

⠀⠀⠀⠀⠀⠀$\huge\underline{ \underline{ \mathbb{ { \blue{ QU{ \pink{ES{ \purple{TION \: : = }}}}} }}}}$

For what value of k the quadratic equation x2-4x+k=0 has real roots

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⠀⠀⠀⠀⠀⠀⠀⠀$\huge\underline{ \underline{ \mathbb{ { \blue{ SO{ \pink{LU{ \purple{TION \: : = }}}}} }}}} </p><p></p><p>$

⠀⠀⠀$\large\underline{ \underline{ \green{ \bold{Given\: equation }}}} = >$

⠀⠀⠀⠀⠀$\bf {x}^{2} - 4x + k$

⠀⠀⠀$\bf \: a = 1 \\ \bf \: b = - 4 \\\bf \: c = k$

⠀⠀⠀⠀⠀⠀⠀⠀⠀$\boxed{ \green{ \bf \: D = {b}^{2} - 4ac}}$

⠀⠀⠀⠀⠀$\mathbb \pink {D= - 4 {}^{2} - 4 \times 1 \times k }$

⠀⠀⠀⠀⠀$\bf \pink{D = 16 - 4k} \\ \\ \bf \pink{D = 4 - k}$

⠀⠀⠀$\bf \pink{D \geqslant 0}$

⠀⠀⠀$\bf \pink{16 - 4k \geqslant 0} \\\bf \pink{ - 4k \geqslant - 16} \\ \\ \bf \pink{k \geqslant \frac{ - 16}{ - 4} }$

⠀⠀⠀⠀⠀⠀$\bf \pink{k \geqslant {\sf{ \cancel{ \dfrac{ - 16}{ - 4}}4}}}$

⠀⠀⠀⠀⠀$\green{ \boxed{ \fbox{ \bf{k <=4}}}}$

Answer 2

${ \bold{ \boxed{ \boxed{ \red{ \huge{ \star \mathbb{ ANSWER \star}}}}}}}$

${ \bold{ \underline{Given} : - }} \\ \\ { \bold{ \blue{ \: \: \: \: . \: \: a \: \: quadratic \: \: equation \: \: {x}^{2} - 4x + k = 0 }}} \\ \\ { \bold{ \blue{ \: \: \: \: . \: \:it's \: \: roots \: \: are \: \: real}}}\\ \\ { \bold{ \underline{ To \: \: find } : - }} \\ { \bold{ \blue{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: value \: \: of \: \: k }}} \\ \\ { \bold{ \boxed{ \boxed{ \green{ \huge \: \mathfrak{ \bigstar \: solution \bigstar}}}}}} \\ \\ { \bold{ \blue{ \: \: \: . \: \: if \: \: roots \: \: are \: \: real \: \: then \: \: discriminant \: \: - }}} \\ \\ { \bold{ \blue{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: { \boxed{ \boxed{D \geqslant 0}}}}}} \\ \\{ \bold{ \blue{ \: \: \: . \: \: but \: \: we \: \: know \: \: that \: \: - }}}\\ \\ { \bold{ \orange{ \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: { \boxed{ D = {b}^{2} - 4ac}}}}} \\ \\ { \bold{ \blue{ \: \: : \implies \:D \geqslant 0}}} \\ \\ { \bold{ \blue{ \: \: : \implies \: {b}^{2} - 4ac \geqslant 0}}} \\ \\ { \bold{ \blue{ \: \: : \implies {( - 4)}^{2} - 4(1)(k) \geqslant 0 }}} \\ \\ { \bold{ \blue{ \: \: : \implies \: \: 16 - 4k \geqslant 0 }}} \\ \\ { \bold{ \blue{ \: \: : \implies \: 4k \leqslant 16}}} \\ \\ { \bold{ \red{ \: \: : \implies { \boxed{ \boxed{\: k \leqslant 4 }}} }}}$