[tex]\mathbb{\red{ \tiny A \scriptsize \: N \small \:S \large \: W \Large \:E \huge \: R}}[/tex]
[tex]\rule{300pt}{0.1pt}[/tex]
Given equations are [tex]\bf x^{2}+p x-4=0 . .(1)[/tex] and [tex]\bf x^{2}+p x+q=0[/tex]...(2)
Since - 4 is a root of (1),
[tex] \text{so \( \bf x=-4 \) must satisfy the equation (1)}[/tex]
[tex] \\ \text{Hence, \( \bf (-4)^{2}+p(-4)-4=0 \)}[/tex]
[tex]\[ \begin{array}{l} \\ \bf \Longrightarrow 16-4 p-4=0 \\ \\ \bf \Longrightarrow 12-4 p=0 \\ \\ \bf \Longrightarrow 4 p=12 \\ \\ \bf \Longrightarrow \: p = \dfrac{12}{4} \\ \\ \boxed{\red{ \bf \Longrightarrow p=3 \: \: \: }} \end{array} \][/tex]
Now, putting p=3 in the equation (2), we get
[tex]\bf \: x^{2}+3 x+q=0[/tex]
[tex] \\ \text{ Discriminant, \( \bf D=b^{2}-4 a c \)}[/tex]
[tex]\[ \begin{array}{l} \\ \bf=(3)^{2}-4(1)(q) \\ \\ \bf=9-4 q \end{array} \][/tex]
Now, (3) has equal roots, so discriminant must be 0
[tex] \text{i.e., \( \bf 9-4 q=0 \)}[/tex]
[tex] \[ \begin{array}{l}\\ \\\bf \Longrightarrow 4 q=9 \\\\ \bf \Longrightarrow q=\dfrac{9}{4} \end{array} \]\\ \\ [/tex]
Hence, the required value of q is [tex]\pmb{\bf \dfrac{9}{4} }[/tex].
[tex]\rule{500pt}{0.1pt}[/tex]
Please scroll from right to left, to see the complete solution.
[tex]\rule{500pt}{0.1mm}[/tex]
[tex]\fcolorbox{magenta}{lavender}{\text{\color{darkcyan} Answerd by ★彡[@ɪᴛꜱ ʏᴏᴜʀᴇ ᴘᴀɢʟᴀ\texttt{02}]彡★}}[/tex]
Copyright © 2024 EHUB.TIPS team's - All rights reserved.
Answers & Comments
[tex]\mathbb{\red{ \tiny A \scriptsize \: N \small \:S \large \: W \Large \:E \huge \: R}}[/tex]
[tex]\rule{300pt}{0.1pt}[/tex]
Given equations are [tex]\bf x^{2}+p x-4=0 . .(1)[/tex] and [tex]\bf x^{2}+p x+q=0[/tex]...(2)
Since - 4 is a root of (1),
[tex] \text{so \( \bf x=-4 \) must satisfy the equation (1)}[/tex]
[tex] \\ \text{Hence, \( \bf (-4)^{2}+p(-4)-4=0 \)}[/tex]
[tex]\[ \begin{array}{l} \\ \bf \Longrightarrow 16-4 p-4=0 \\ \\ \bf \Longrightarrow 12-4 p=0 \\ \\ \bf \Longrightarrow 4 p=12 \\ \\ \bf \Longrightarrow \: p = \dfrac{12}{4} \\ \\ \boxed{\red{ \bf \Longrightarrow p=3 \: \: \: }} \end{array} \][/tex]
Now, putting p=3 in the equation (2), we get
[tex]\bf \: x^{2}+3 x+q=0[/tex]
[tex] \\ \text{ Discriminant, \( \bf D=b^{2}-4 a c \)}[/tex]
[tex]\[ \begin{array}{l} \\ \bf=(3)^{2}-4(1)(q) \\ \\ \bf=9-4 q \end{array} \][/tex]
Now, (3) has equal roots, so discriminant must be 0
[tex] \text{i.e., \( \bf 9-4 q=0 \)}[/tex]
[tex] \[ \begin{array}{l}\\ \\\bf \Longrightarrow 4 q=9 \\\\ \bf \Longrightarrow q=\dfrac{9}{4} \end{array} \]\\ \\ [/tex]
Hence, the required value of q is [tex]\pmb{\bf \dfrac{9}{4} }[/tex].
[tex]\rule{500pt}{0.1pt}[/tex]
η◎†℮ :-
Please scroll from right to left, to see the complete solution.
[tex]\rule{500pt}{0.1mm}[/tex]
[tex]\fcolorbox{magenta}{lavender}{\text{\color{darkcyan} Answerd by ★彡[@ɪᴛꜱ ʏᴏᴜʀᴇ ᴘᴀɢʟᴀ\texttt{02}]彡★}}[/tex]