Question

If the roots of the equation are equal then find the value of b^2:(a^2+b^2)x- 2b(a+c)X+(b^2+c^2)=0

Answer 1

Solution:

If the equation has equal roots then it's Discriminate D = 0

$D =0 \\ \\ {b}^{2} - 4ac = 0 \\ \\ 4 {b}^{2} ( {a + c)}^{2} - 4( {a}^{2} + {b}^{2}) ( {b}^{2} + {c}^{2} ) = 0 \\ \\ 4 {b}^{2} ( {a}^{2} + {c}^{2} + 2ac) - 4( {a}^{2} {b}^{2} + {a}^{2} {c}^{2} + {b}^{4} + {b}^{2} {c}^{2} ) = 0 \\ \\ 4 {b}^{2} {a}^{2} + 4 {b}^{2} {c}^{2} + 8 {b}^{2}ac - 4 {a}^{2} {b}^{2} - 4 {a}^{2} {c}^{2} - 4 {b}^{4} - 4{b}^{2} {c}^{2} = 0 \\ \\ - {b}^{4} - {a}^{2} {c}^{2} + 4a {b}^{2} c = 0 \\ \\ {b}^{4} + {a}^{2} {c}^{2} - 2a {b}^{2} c = 0$
${( {b}^{2} })^{2} - 2(ac)( {b}^{2} ) + {(ac)}^{2} = 0 \\ \\ {( {b}^{2} - ac) }^{2} = 0 \\ \\ {b}^{2} - ac = 0 \\ \\ {b}^{2} = ac$
Hope it helps you.