Nd maximum value of x1^2+x2^2 where x1 and x2 are roots of quadratic e

1.	Nd maximum value of x1^2+x2^2 where x1 and x2 are roots of quadratic equation x^2-kx+(k^2+7k+15)maths kings iska ans with explaination send kijieagar ans Sahi to me aapko follow karunga nahito aap muje follow karana or ha ans 19 Nahi he
Answer» GIVEN :– • Quadratic equation x² - kx + (k² + 7k + 15) = 0. • Roots of quadratic equation $\implies \bf \:\: x_1 \: and \: x_2 \:\:$ TO FIND :– • Maximum value of $\bf \:\: x_1^{2} +x_2^{2} = ?\:\:$ SOLUTION :– • We should WRITE this as – $\\\implies\bf x_1^{2} +x_2^{2} =(x_1 + x_2)^{2} - 2(x_1)(x_2)\: \: \: \: \:\: - - - -<klux>EQ</klux>.(1)$ • We KNOW that – $\\\implies\bf Sum \: \: of \: \: roots = - \dfrac{Coefficient \: \: of \: \: x}{Coefficient \: \:of \: \: {x}^{2} }\:\:$ $\\\implies\bf x_1 + x_2= - \dfrac{( - k)}{1}\:\:$ $\\\implies\bf x_1 + x_2=k \: \: \: \: \:\: - - - -eq.(2)$ • And – $\\\implies\bf Product \: \: of \: \: roots = \dfrac{Constent \: \: term}{Coefficient \: \:of \: \: {x}^{2} }\:\:$ $\\\implies\bf (x_1)( x_2)= \dfrac{( {k}^{2} + 7k + 15)}{1}\:\:$ $\\\implies\bf (x_1)( x_2)={k}^{2} + 7k + 15 \: \: \: \: \:\: - - - -eq.(3)$ • Put the VALUES from eq.(3) & eq.(2) in eq.(1) $\\\implies\bf x_1^{2} +x_2^{2} =(k)^{2} - 2({k}^{2} + 7k + 15)$ • Let – $\\\implies\bf x_1^{2} +x_2^{2} =P$ • So that – $\\\implies\bf P =k^{2} - 2{k}^{2} - 14k - 30$ $\\\implies\bf P = - {k}^{2} - 14k - 30$ • Now Differentiate with respect to 'k' – $\\\implies\bf \dfrac{dP}{dk} = -2k- 14$ • For Maximum/Minimum – $\\\implies\bf \dfrac{dP}{dk} = 0$ $\\\implies\bf -2k- 14= 0$ $\\\implies\bf 2k + 14= 0$ $\\\implies\bf 2k = - 14$ $\\\implies \large{ \boxed{\bf k = -7}}$ • Now 'P' – $\\\implies\bf P_{max} = - {(- 7)}^{2} - 14( - 7) - 30$ $\\\implies\bf P_{max} = -49 + 98 - 30$ $\\\implies\bf P_{max} = 98 - 79$ $\\\implies \large{ \boxed{\bf P_{max} =19}}$ ▪︎ Hence , Maximum value of $\bf \:x_1^{2} +x_2^{2} \:$ is 19.