**Check If the Given Expression Form a Quadratic Equation :**

The general form of any quadratic equation is

ax^{2} + bx + c = 0

Here a ≠ 0

If the expression can be expressed in the general form, we may say that the given expression is quadratic equation.

(i) (x + 1)² = 2 (x – 3)

**Solution :**

(x + 1)² = 2 (x – 3) can be rewritten as

(a + b)² = a^{2} + 2ab + b^{2}

x²+ 2 x + 1 = 2 x – 6

x²+ 2 x – 2 x + 1 + 6 = 0

x² + 7 = 0

This exactly matches the general form of quadratic equation. So, the given equation is quadratic equation.

(ii) x² – 2 x = (-2) (3 – x)

**Solution :**

x² – 2 x = - 6 + 2 x

x² – 2 x – 2 x + 6 = 0

x² – 4 x + 6 = 0

This exactly matches the general form of quadratic equation.

(iii) (x - 2) (x + 1) = (x - 1) (x + 3)

**Solution :**

x² + 1x – 2x - 2 = x² + 3x - 1x - 3

x² - 1x - 2 = x² + 2x - 3

x² - x²- 1x - 2x - 2 + 3 = 0

- 3x + 1 = 0

It does not match with the general form of quadratic equation. So the given equation is not a quadratic equation.

(iv) (x - 3) (2x + 1) = x (x + 5)

**Solution :**

2x² + 1x – 6x - 3 = x² + 5x

2x² – 5x - 3 = x² + 5x

2x²- x² - 5x – 5x - 3 = 0

x² - 10x - 3 = 0

This exactly matches the general form of quadratic equation. So the given equation is a quadratic equation.

(v) (2 x - 1) (x - 3) = (x + 5) (x - 1)

**Solution :**

2 x² - 6 x – x + 3 = x² - 1 x - 5 x - 5

2 x² - 7 x + 3 = x² - 6 x - 5

2 x²- x²- 7 x + 6 x + 3 + 5 = 0

x² - 1 x + 8 = 0

This exactly matches the general form of quadratic equation. So, the given equation is a quadratic equation.

(vi) x² + 3 x + 1 = (x - 2)²

**Solution :**

x² + 3 x + 1 = x² - 2 x (2) + 2²

x² + 3 x + 1 = x² - 4 x + 4

x²- x² + 3 x + 4 x + 1 - 4 = 0

7 x - 3 = 0

This does not match the general form of quadratic equation. So, the given equation is not a quadratic equation.

(vii) (x + 2)³ = 2 x (x² - 1)

**Solution :**

x³ + 3 (x²) (2) + 3 (x) (2)² + 2³ = 2 x³ - 2 x

x³ + 6 x² + 12 x + 8 = 2 x³ - 2 x

x³ - 2 x³ + 6 x² + 12 x + 2 x + 8 = 0

- x³ + 6 x² + 12 x + 2 x + 8 = 0

This does not exactly match the general form of quadratic equation. So the given equation is not a quadratic equation.

(vii) x³ - 4 x² - x + 1 = (x - 2)³

**Solution :**

x³ - 4 x² - x + 1 = x³ - 3 (x²) (2) + 3 (x) (2)² - 2³

x³ - 4 x² - x + 1 = x³ - 6 x² + 12 x - 8

x³- x³ - 4 x² + 6 x² - x - 12 x + 1 + 9 = 0

2 x² - 13 x + 10 = 0

This exactly matches with the general form of quadratic equation. So the given equation is a quadratic equation.

After having gone through the stuff and examples, we hope that the students would have understood, check if the given expression form a quadratic equation.

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