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  • Find the roots of the following quadratic equations by the factorisation method (i)2x^{2}+\frac{5}{3}x-2=0 (ii)\frac{2}{5}x^{2}-x-\frac{3}{5}=0 (iii)3 \sqrt{2}x^{2}-5x-\sqrt{2}=0

Find the roots of the following quadratic equations by the factorisation method

(i)2x^{2}+\frac{5}{3}x-2=0

(ii)\frac{2}{5}x^{2}-x-\frac{3}{5}=0

(iii)3 \sqrt{2}x^{2}-5x-\sqrt{2}=0

(iv)3x^{2}+5\sqrt{5}x-10=0

(v)21x^{2}-2x+\frac{1}{21}=0

Answers (1)

(ii) 3,-\frac{1}{2}

Solution

 \\\frac{2}{5}x^{2}-x-\frac{3}{5}=0\\ \frac{2x^{2}-5x-3}{5}=0\\ 2x^{2}-5x-3=0\\ 2x^{2}-6x+x-3=0\\ 2x(x-3)+1(x-3)=0\\ (x-3)(2x+1)=0\\ x-3=0\;\;\;\;\;\;\;\;\;\;\;\;\; 2x+1=0\\ x=3\;\;\;\;\;\;\;\;\;\;\;\;\; 2x=-1 \;\;\;\;x=\frac{-1}{2}\\

Hence \frac{-3}{2},\frac{2}{3} are the roots of the equation.

(i) \frac{-3}{2},\frac{2}{3}

Solution

 2x^{2}+\frac{5}{3}x-2=0\\ \frac{6x^{2}+5x-6}{3}=0\\ 6x^{2}+5x-6=0\\ 6x^{2}+9x-4x-6=0\\ 3x(2x+3)-2(2x+3)=0\\ (2x+3)(3x-2)=0\\ 2x+3=0\;\;\;\;\;\;\;\;\;\;\;\;\; 3x-2=0\\ 2x=-3\;\;\;\;\;\;\;\;\;\;\;\;\; 3x=2\\ x=\frac{-3}{2}\;\;\;\;\;\;\;\;\;\;\;\;\; x=\frac{2}{3}\\

Hence 3,-\frac{1}{2} are the roots of the equation.

(iii) \sqrt{2},-\frac{1}{3\sqrt{2}}

Solution

   3\sqrt{2}x^{2}-5x-\sqrt{2}=0\\ 3\sqrt{2}x^{2}-6x+x-\sqrt{2}=0\\ 3\sqrt{2}x(x-\sqrt{2})+1(x-\sqrt{2})=0\\ (x-\sqrt{2})(3\sqrt{2}x+1)=0\\ 3\sqrt{2}x+1=0 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x-\sqrt{2}=0\\ 3\sqrt{2}x=-1 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x=\sqrt{2}\\ x=\frac{-1}{3\sqrt{2}}

Hence \sqrt{2},-\frac{1}{3\sqrt{2}} are root of the equation 3\sqrt{2}x^{2}-5x-\sqrt{2}=0\\

(iv) -2\sqrt{5},-\frac{\sqrt{5}}{3}

Solution

   3x^{2}+5\sqrt{5}x-10=0\\ 3x^{2}+6\sqrt{5}x-\sqrt{5}x-10=0\\ 3x(x+2\sqrt{5})-\sqrt{5}(x+2\sqrt{5})=0\\ (3x-\sqrt{5})(x+2\sqrt{5})=0\\ 3x-\sqrt{5}=5 \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x+2\sqrt{5}=0\\ 3x=\sqrt{5} \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x=-2\sqrt{5}\\ x=\frac{\sqrt{5}}{3}

Hence -2\sqrt{5},-\frac{\sqrt{5}}{3} are root of the equation 3x^{2}+5\sqrt{5}x-10=0

(v) \frac{1}{21}

solution

21x^{2}-2x+\frac{1}{21}=0\\ 21x^{2}-x-x+\frac{1}{21}=0\\ x(21x-1)-\frac{1}{21}(21x-1)=0\\ (21x-1)\left ( x-\frac{1}{21} \right )=0\\ 21-x=0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x-\frac{1}{21}=0\\ 21x=1\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x=\frac{1}{21}\\ x=\frac{1}{21}

Hence \frac{1}{21} are the roots of the equation 21x^{2}-2x+\frac{1}{21}=0

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