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Which of the following is a quadratic equation?

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

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

2 x^{2} + 3 = (5 + x) (2 x - 3)

None of these

answer is B.

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Detailed Solution

To find the quadratic equation.
We know that, standard form of quadratic equation is,

a x^{2} + bx + c = 0

.
In option (1),

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

⟹ x^{2} + 1 = 4 - 4 x + x^{2} + 3

⟹ x^{2} + 1 - 4 + 4 x - x^{2} - 3 = 0

⟹ 4 x - 6 = 0

This equation is of degree 1 and as we know, a quadratic equation is of 2 degree.
Therefore, it is not a quadratic equation.
In option (2),

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

{⟹ x}^{3} - x^{2} = x^{3} + 3 x^{2} + 3 x + 1 \{∴ {(a + b)}^{3} = a^{3} + 3 a^{2} b + 3 a b^{2} + b^{3}\}

⟹ x^{3} - x^{2} - x^{3} - 3 x^{2} - 3 x - 1 = 0

⟹ x^{3} - x^{3} - x^{2} - 3 x^{2} - 3 x - 1 = 0

⟹ - 4 x^{2} - 4 x - 1 = 0

After multiplying the whole equation by the minus sign, we get,

4 x^{2} + 4 x + 1 = 0

This equation is quadratic because it is of degree 2 and is in the standard form

a x^{2} + bx + c = 0

.
Therefore, it is a quadratic equation.
Now, let’s check for the third option,

2 x^{2} + 3 = (5 + x) (2 x - 3)

⟹ 2 x^{2} + 3 = 5 (2 x - 3) + x (2 x - 3)

⟹ 2 x^{2} + 3 = 10 x - 15 + 2 x^{2} - 3 x

⟹ 2 x^{2} + 3 = 2 x^{2} + 7 x - 15

⟹ 2 x^{2} - 2 x^{2} - 7 x + 15 + 3 = 0

⟹ - 7 x + 18 = 0

After multiplying the whole equation by the minus sign, we will get,

7 x - 18 = 0

This equation is of degree 1 and as we know, a quadratic equation is of 2 degree.
Therefore, it is not a quadratic equation.
Hence, the correct option is 2

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