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  • Which of the following equations has two distinct real roots? (A)2x^{2}-3\sqrt{2}x+\frac{9}{4}=0\\ (B)x^{2}+x-5=0\\ (C)x^{2}+3x+2\sqrt{2}=0\\ (D)5x^{2}-3x+1=0

Which of the following equations has two distinct real roots?

\\(A)2x^{2}-3\sqrt{2}x+\frac{9}{4}=0\\ (B)x^{2}+x-5=0\\ (C)x^{2}+3x+2\sqrt{2}=0\\ (D)5x^{2}-3x+1=0

Answers (1)

We know that if roots are distinct and real, then b^{2}-4ac>0

 (A)2x^{2}-3\sqrt{2}x+\frac{9}{4}=0

\text{compare with } ax^{2}+bx+c=0 \text{where} a \neq 0

 a=2, b=-3\sqrt{2},c=\frac{9}{4}
  \\b^{2}-4ac=\left ( -\frac{3}{\sqrt{2}} \right )^{2}-4(2)\left ( \frac{9}{4} \right )\\ =9 \times 2-18\\ =18-18=0\\ b^{2}-4ac=0\text{(two equal real roots)}    

  (B)\ x^{2}+x-5=0
 b^{2}-4ac=(1)^{2}-4(1)(-5)\\ =1+20 =21\\ b^{2}-4ac>0\text{(two distinct real roots) }                              

  (C)\ x^{2}+3x+2\sqrt{2}=0

           
  \\b^{2}-4ac=(3 )^2-4(1) ( 2\sqrt{2} )\\ =9 -8\sqrt{2}\\ =9-11.3=-2.3\\ b^{2}-4ac<0\text{(no real roots)}  

(D)\ 5x^{2}-3x+1=0        
  \\b^{2}-4ac=\left (-3 \right )^{2}-4(5)\left ( 1 \right )\\ =9 -20\\ =-11\\ b^{2}-4ac<0\text{(no real roots)}  

Here only option (B)  have two distinct real roots.

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