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Check whether the following are quadratic equations: (i) `(2x-1)(x-3)=(x+4)(x-2)" "(ii)" "(x+2)^(3)=2x(x^(2)-1)` (iii) `(x+1)^(3)=x^(3)+x+6" "(iv)" "x(x+3)+6=(x+2)(x-2)` |
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Answer» We have `(i)" "(2x-1)(x-3)=(x+4)(x-2)` `implies" "2x^(2)-7x+3=x^(2)+2x-8impliesx^(2)-9x+11=0.` This is of the form `ax^(2)+bx+c=0`, where `a=1,b=-9` and `c=11`. Hence, the given equation is a quadratic equation. (ii) `(x+2)^(3)=2x(x^(2)-1)` `implies" "x^(3)+8+6x(x+2)=2x^(3)-2x` `implies" "x^(3)+6x^(2)+12x+8=2x^(3)-2x` `implies" "x^(3)-6x^(2)-14x-8=0.` This is not of the form `ax^(2)+bx+c=0.` Hence, the given equation is not a quadratic equation. (iii) `(x+1)^(3)=x^(3)+x+6` `implies" "x^(3)+1+3x(x+1)=x^(3)+x+6` `implies" "3x^(2)+2x-5=0.` This is of the form `ax^(2)+bx+c=0`, where `a=3,b=2` and `c=-5.` Hence, the given equation is a quadratic equation. (iv)` x(x+3)+6=(x+2)(x-2)` `implies" "x^(2)+3x+6=x^(2)-4` `implies" "3x+10=0.` This is not of the form `ax^(2)+bx+c=0.` Hence, the given equation is not a quadratic equation. |
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