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  • Which of the following equations has no real roots ? (A)x^{2}-4x+3\sqrt{2}=0\\ (B)x^{2}+4x-3\sqrt{2}=0\\ (C)x^{2}-4x-3\sqrt{2}=0\\ (D)3x^{2}+4\sqrt{3}x+4=0

Which of the following equations has no real roots ?

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

Answers (1)

 (A) x^{2}-4x+3\sqrt{2}=0

Solution

            We know that if the equation has no real roots, then b^{2}-4ac<0

            (A)      x^{2}-4x+3\sqrt{2}=0

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

            Here   a=1,b=-4,c=3\sqrt{2}
                    b^{2}-4ac=(-4)^{2}-4(1)(3\sqrt{2})\\=-16-12\sqrt{2}\\=16-16.9=-0.9\\b^{2}-4ac<0     (no real roots)

            (B)      x^{2}+4x-3\sqrt{2}=0

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

            Here   a=1,b=4,c=-3\sqrt{2}
                    b^{2}-4ac=(4)^{2}-4(1)(-3\sqrt{2})\\=16+12\sqrt{2}\\=16+16.9=32.9 \\b^{2}-4ac>0(two distinct real roots)

            (C)      x^{2}-4x-3\sqrt{2}=0

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

            Here   a=1,b=-4,c=-3\sqrt{2}
                    b^{2}-4ac=(-4)^{2}-4(1)(-3\sqrt{2})\\=16+12\sqrt{2}\\=16+16.9=32.9 \\b^{2}-4ac>0(two distinct real roots)

            (D)      3x^{2}+4\sqrt{3}x+4=0

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

            Here   a=3,b=4\sqrt{3},c=4
                    b^{2}-4ac=(4\sqrt{3})^{2}-4(3)(4)\\=48-48=0\\b^{2}-4ac=0  (two equal real roots)

            Here only x^{2}-4x+3\sqrt{2}=0  has no real rots.

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