Questions & Answers

Question

Answers

$

(a)\,{\left( {x - 2} \right)^2} + 1 = 2x - 3 \\

(b){\text{ }}x\left( {x + 1} \right) + 8 = \left( {x + 2} \right)\left( {x - 2} \right) \\

(c){\text{ }}x\left( {2x + 3} \right) = {x^2} + 1 \\

(d){\text{ }}{\left( {x + 2} \right)^3} = {x^3} - 4 \\

$

Answer
Verified

Hint – A quadratic equation is one which has the general form of $a{x^2} + bx + c = 0$ where $a \ne 0$. Simplify the given options step by step to check if it can be reduced to this general form or not.

Complete step-by-step answer:

As we know that in a quadratic the highest power of a variable is 2.

The general quadratic equation is $a{x^2} + bx + c = 0$

Now check options one by one we have,

$\left( i \right){\text{ }}{\left( {x - 2} \right)^2} + 1 = 2x - 3$

Now simplify this equation we have according to the formula${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$.

$ \Rightarrow {x^2} + 4 - 4x + 1 = 2x - 3$

$ \Rightarrow {x^2} - 6x + 8 = 0$

So this represents a quadratic equation.

Now check option (ii)

$\left( {ii} \right){\text{ }}x\left( {x + 1} \right) + 8 = \left( {x + 2} \right)\left( {x - 2} \right)$

Now simplify this equation we have,

$ \Rightarrow {x^2} + x + 8 = {x^2} - 4$

$ \Rightarrow x + 12 = 0$

So this does not represent the quadratic equation.

Now check option (iii)

$\left( {iii} \right){\text{ }}x\left( {2x + 3} \right) = {x^2} + 1$

Now simplify this equation we have,

$ \Rightarrow 2{x^2} + 3x = {x^2} + 1$

$ \Rightarrow {x^2} + 3x - 1 = 0$

So this also represents a quadratic equation.

Now check option (iv)

$\left( {iv} \right){\text{ }}{\left( {x + 2} \right)^3} = {x^3} - 4$

Now simplify this equation we have according to the formula${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3a{b^2} + 3{a^2}b$.

$ \Rightarrow {x^3} + 8 + 6{x^2} + 12x = {x^3} - 4$

$ \Rightarrow 6{x^2} + 12x + 12 = 0$

Now divide by 6 we have,

$ \Rightarrow {x^2} + 2x + 2 = 0$

So this also represents a quadratic equation.

So from amongst options only option (B) does not represent the quadratic equation.

So option (B) is the required answer.

Note – In this question a basic algebraic identity of ${\left( {a + b} \right)^3}$is also used, we know that on solving this forms a cubic equation so directly we could have said that it won’t be quadratic . The trick to solve these types of problems is check for the highest power of x in the equation if it’s 2 that means the equation is quadratic else not.

Complete step-by-step answer:

As we know that in a quadratic the highest power of a variable is 2.

The general quadratic equation is $a{x^2} + bx + c = 0$

Now check options one by one we have,

$\left( i \right){\text{ }}{\left( {x - 2} \right)^2} + 1 = 2x - 3$

Now simplify this equation we have according to the formula${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$.

$ \Rightarrow {x^2} + 4 - 4x + 1 = 2x - 3$

$ \Rightarrow {x^2} - 6x + 8 = 0$

So this represents a quadratic equation.

Now check option (ii)

$\left( {ii} \right){\text{ }}x\left( {x + 1} \right) + 8 = \left( {x + 2} \right)\left( {x - 2} \right)$

Now simplify this equation we have,

$ \Rightarrow {x^2} + x + 8 = {x^2} - 4$

$ \Rightarrow x + 12 = 0$

So this does not represent the quadratic equation.

Now check option (iii)

$\left( {iii} \right){\text{ }}x\left( {2x + 3} \right) = {x^2} + 1$

Now simplify this equation we have,

$ \Rightarrow 2{x^2} + 3x = {x^2} + 1$

$ \Rightarrow {x^2} + 3x - 1 = 0$

So this also represents a quadratic equation.

Now check option (iv)

$\left( {iv} \right){\text{ }}{\left( {x + 2} \right)^3} = {x^3} - 4$

Now simplify this equation we have according to the formula${\left( {a + b} \right)^3} = {a^3} + {b^3} + 3a{b^2} + 3{a^2}b$.

$ \Rightarrow {x^3} + 8 + 6{x^2} + 12x = {x^3} - 4$

$ \Rightarrow 6{x^2} + 12x + 12 = 0$

Now divide by 6 we have,

$ \Rightarrow {x^2} + 2x + 2 = 0$

So this also represents a quadratic equation.

So from amongst options only option (B) does not represent the quadratic equation.

So option (B) is the required answer.

Note – In this question a basic algebraic identity of ${\left( {a + b} \right)^3}$is also used, we know that on solving this forms a cubic equation so directly we could have said that it won’t be quadratic . The trick to solve these types of problems is check for the highest power of x in the equation if it’s 2 that means the equation is quadratic else not.

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