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Question
Find the values of k for which the roots are real and equal in each of the following equations.
(i)kx2+4x+1=0(ii)kx2−2√5x+4=0(iii)3x2−5x+2k=0(iv)4x2+kx+9=0(v)2kx2−40x+25=0(vi)9x2−24x+k=0(vii)4x2−2kx+1=0(viii)x2−2(5+2k)x+3(7+10k)=0(ix)(3k+1)x2+2(k+1)x+k=0(x)kx2+kx+1=−4x2−x(xi)(k+1)x2+2(k+3)x(k+8)=0(xii)x2−2kx+7k−12=0(xiii)(k+1)x2−2(3k+1)x+8k+1=0(xiv)5x2−4x+2+k(4x21−2x−1)=0(xv)(4−k)x2+(2k+4)x+(8k+1)=0(xvi)(2k+1)x2+2(k+3)x+(k+5)=0(xvii)4x2−2(k+1)x+(k+4)=0(xviii)4x2−2(k+1)x+(k+1)=0
Find the values of k for which the roots are real and equal in each of the following equations.
(i)kx2+4x+1=0(ii)kx2−2√5x+4=0(iii)3x2−5x+2k=0(iv)4x2+kx+9=0(v)2kx2−40x+25=0(vi)9x2−24x+k=0(vii)4x2−2kx+1=0(viii)x2−2(5+2k)x+3(7+10k)=0(ix)(3k+1)x2+2(k+1)x+k=0(x)kx2+kx+1=−4x2−x(xi)(k+1)x2+2(k+3)x(k+8)=0(xii)x2−2kx+7k−12=0(xiii)(k+1)x2−2(3k+1)x+8k+1=0(xiv)5x2−4x+2+k(4x21−2x−1)=0(xv)(4−k)x2+(2k+4)x+(8k+1)=0(xvi)(2k+1)x2+2(k+3)x+(k+5)=0(xvii)4x2−2(k+1)x+(k+4)=0(xviii)4x2−2(k+1)x+(k+1)=0
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Solution
(i)k=4(ii)k=54(iii)k=2524(iv)k=±12(v)k=8(vik=16)(vii)k=±43(viii)k=2,12(ix)k=−12,1(x)k=5,−3(xi)k=13(xii)k=4,3(xiii)k=0,3(xiv)k=−65,1(xv)k=0,3(xvi)k=−5±√412(xvii)k=−3,−7(xviii)k=−1,3
(i)k=4(ii)k=54(iii)k=2524(iv)k=±12(v)k=8(vik=16)(vii)k=±43(viii)k=2,12(ix)k=−12,1(x)k=5,−3(xi)k=13(xii)k=4,3(xiii)k=0,3(xiv)k=−65,1(xv)k=0,3(xvi)k=−5±√412(xvii)k=−3,−7(xviii)k=−1,3
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