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  • State whether the following quadratic equations have two distinct real roots. Justify your answer. (i)x^{2}-3x+4=0 (ii)2x^{2}+x-1=0 (iii)2x^{2}-6x+\frac{9}{2}=0

State whether the following quadratic equations have two distinct real roots. Justify your answer.

(i)x^{2}-3x+4=0         

(ii)2x^{2}+x-1=0

(iii)2x^{2}-6x+\frac{9}{2}=0

(iv)3x^{2}-4x+1=0

(v)(x+4)^{2}-8x=0

(vi)(x-\sqrt{2})^{2}-2(x+1)=0

(vii)\sqrt{2}x^{2}-\frac{3}{\sqrt{2}}x+\frac{1}{2}=0

(viii)x(1-x)-2=0

(ix)(x-1)(x+2)+2=0

(x)(x+1)(x-2)+x=0

Answers (1)

(i)     Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0  Where a, b and c are real numbers with a\neq 0

            The given equation is x^{2}-3x+4=0

            Here a=1,b=-3,c=4
                      D=b^{2}-4ac\\ =(-3)^{2}-4(1)(4)\\ =9-16=-7\\ b^{2}-4ac<0

     Hence the equation has no real roots.

(ii)   Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0           Where a, b and c are real numbers with a\neq 0

            The given equation is 2x^{2}+x-1=0

            Here a=2,b=-1,c=-1
                      D=b^{2}-4ac\\ =(1)^{2}-4(2)(-1)\\ =1+8=9\\ b^{2}-4ac>0

            Here for the equation 2x^{2}+x-1=0,  b^{2}-4ac>0

            Hence it is a quadratic equation with two distinct real roots

(iii)   Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0           Where a, b and c are real numbers with a\neq 0

            The given equation is 2x^{2}-6x+\frac{9}{2}=0

            Here a= 2,b=-6,c=\frac{9}{2}

                      D=b^{2}-4ac\\ =(-6)^{2}-4(2)(\frac{9}{4})\\ =36-36=0\\ b^{2}-4ac=0

        here b^{2}-4ac=0

        Hence the equation have two real and equal roots.

(iv)    Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0           Where a, b and c are real numbers with a\neq 0

            The given equation is 3x^{2}-4x+1=0

            Here a=3,b=-4,c=1
                      D=b^{2}-4ac\\ =(-4)^{2}-4(3)(1)\\ =16-12=4\\ b^{2}-4ac>0

            Here  b^{2}-4ac>0

            Hence the equation has distinct and real roots.

(v) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0           Where a, b and c are real numbers with a\neq 0

            The given equation is (x+4)^{2}-8x=0

            (x)^{2}+(4)^{2}+2(x)(4)-8x=0                                              \left (using (a+b)^{2}=(a)^{2}+(b)^{2}+2ab \right )

            (x)^{2}+16+8x-8x=0\\ x^{2}+16=0

            a=1,b=0,c=16\\ =(0)^{2}-4(1)(16)\\ =-64\\ b^{2}-4ac<0

            Here  b^{2}-4ac<0

            Hence the equation has no real roots.

(vi) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0           Where a, b and c are real numbers with a\neq 0

            The given equation is (x-\sqrt{2})^{2}-2(x+1)=0

            (x)^{2}+(\sqrt{2})^{2}+2(x)(\sqrt{2})-2x-2=0                        \left (using (a+b)^{2}=(a)^{2}+(b)^{2}+2ab \right )

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

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

            a=1,b=-(2\sqrt{2}+2),c=0\\b^{2}-4ac =(-(2\sqrt{2}+2))^{2}-4(1)(0)\\ =(2\sqrt{2}+2)^{2}-0

            =(2\sqrt{2})^{2}+(2)^{2}+2(2\sqrt{2})(2)                            \left (using (a+b)^{2}=(a)^{2}+(b)^{2}+2ab \right )

            =8+4+8\sqrt{2}\\ =12+8\sqrt{2}\\

            b^{2}-4ac>0 

            Here  b^{2}-4ac>0

            Hence the equation has two distinct real roots.

(vii)    Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0           Where a, b and c are real numbers with a\neq 0

            The given equation is \sqrt{2}x^{2}-\frac{3}{\sqrt{2}}x+\frac{1}{2}=0

            Here a=\sqrt{2},b=\frac{3}{\sqrt{2}},c=\frac{1}{2}
                      b^{2}-4ac =\left (- \frac{3}{\sqrt{2}} \right )^{2}-4(\sqrt{2})\left (\frac{1}{2} \right )\\ =\frac{9}{2}-2\sqrt{2}\\=4.5-3.8=1.7\\ b^{2}-4ac>0

            Here  b^{2}-4ac>0

            Hence the equation has distinct real roots.

(viii) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0           Where a, b and c are real numbers with a\neq 0

            The given equation is x(1-x)-2=0

            x-(x)^{2}-2=0\\ -x^{2}+x-2=0

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

            a=-1,b=1,c=-2\\ b^{2}-4ac=(1)^{2}-4(-1)(-2)\\1-8=-7\\ b^{2}-4ac<0

            Here  b^{2}-4ac<0

            Hence the equation has no real roots.

(ix) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0           Where a, b and c are real numbers with a\neq 0

            The given equation is (x-1)(x+2)+2=0

            x(x+2)-1(x+2)+2=0\\ x^{2}+2x-x-2+2=0\\ x^{2}+x=0\\

            a=1,b=1,c=0\\b^{2}-4ac =(1)^{2}-4(1)(0)\\ =1

            b^{2}-4ac>0 

            Here  b^{2}-4ac>0

            Hence the equation has two distinct real roots.

(x) Quadratic equation : A quadratic equation in x is an equation that can be written in the standard form ax^{2}+bx+c=0           Where a, b and c are real numbers with a\neq 0

            The given equation is (x+1)(x-2)+x=0

            x(x-2)+1(x-2)+x=0\\ x^{2}-2x+x-2+x=0\\ x^{2}-2x+2x-2=0\\x^{2}-2=0

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

            a=1,b=0,c=-2\\b^{2}-4ac =(0)^{2}-4(1)(-2)\\ =8

            b^{2}-4ac>0 

            Here  b^{2}-4ac>0

            Hence the equation has two distinct real roots.

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