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If AD is not equal to BC then prove that the equation (a^2+b^2)x^2+2(ac+bd)x+(c^2+d^2)=o has no real roots.

Posted by Manish Jat 6 years, 9 months ago

CBSE > Class 10 > Mathematics

1 answers

Sia ? 6 years, 9 months ago

Given, (a²+ b²)x²+ 2(ac +bd)x + (c²+ d²) = 0
The given equation is of the form {tex}Ax^2+Bx+C=0{/tex}
{tex}A=a^2+b^2,\ \ B=2(ac+bd),\ \ C=c^2+d^2{/tex}
D = b² - 4ac
= {tex}[(2(ac + bd))^2 - 4 (a^2 + b^2) (c^2 + d^2)]{/tex}
= {tex}[4(a^2c^2 + 2abcd + b^2d^2] - 4[a^2c^2 + a^2d^2 + b^2c^2 + b^2d^2]{/tex}
= 4a²c²+ 8abcd + 4b²d² - 4a²c²- 4a²d²- 4b²c²- 4b²d²
= 8abcd - 4a²d² - 4b²c²
= -4[a²d² + b²c² - 2abcd]
= -4[ad - bc]² < 0
Hence, the equation has no real roots.

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CBSE > Class 10 > Mathematics