### Question 1. Write the following quadratic equations in the standard form and point out pure quadratic equations.

$ \pmb{ (i). \;\;\;\;\; (x + 7) (x – 3) = -7} $

### Solution:

The given quadratic equation is

$$ (x + 7) (x – 3) = -7 \;\;\;\;\; (i) $$

Multiplying the expressions $ (x + 7) $ and $ (x – 3) $ in equation (i) with each other, we have

$$ x^2 + 7x – 3x – 21 = -7 $$

$$ \Rightarrow x^2 + 4x – 21 = -7 $$

Now adding 7 on both sides of the above equation and get

$$ x^2 + 4x – 21 + 7 = -7 + 7 $$

$$ \Rightarrow x^2 + 4x – 14 = 0 \;\;\;\;\; (ii) $$

Equation (ii) is in the standard form of the quadratic equation but it is not a pure quadratic equation because b ≠ 0.

$ \pmb{ (ii). \;\;\;\;\; {x^2 + 4 \over 3} – {x \over 7} = 1} $

### Solution:

The given quadratic equation is

$$ {x^2 + 4 \over 3} – {x \over 7} = 1 \;\;\;\;\; (i) $$

Taking LCM (Least Common Multiple) of 3 and 7 that is 21, now multiply both sides of the equation (i) by 21, we have

$$ 21({x^2 + 4 \over 3}) – 21({x \over 7}) = 21(1) $$

$$ \Rightarrow 7(x^2 + 4) – 3x = 21 $$

$$ \Rightarrow 7x^2 + 28 – 3x = 21 $$

$$ \Rightarrow 7x^2 – 3x + 28 = 21 $$

Now subtracting 21 from both sides of the above equation and get

$$ 7x^2 – 3x + 28 – 21 = 21 – 21 $$

$$ \Rightarrow 7x^2 – 3x + 7 = 0 \;\;\;\;\; (ii) $$

Equation (ii) is in the standard form of the quadratic equation but it is not a pure quadratic equation because b ≠ 0.

$ \pmb{ (iii). \;\;\;\;\; {x \over x + 1} + {x + 1 \over x} = 6 } $

### Solution:

The given quadratic equation is

$$ {x \over x + 1} + {x + 1 \over x} = 6 \;\;\;\;\; (i) $$

By solving the expression $ {x \over x + 1} + {x + 1 \over x} $ in equation (i), we have

$$ {x^2 + x^2 + 2x + 1 \over x(x + 1)} = 6 $$

$$ \Rightarrow {2x^2 + 2x + 1 \over x^2 + x} = 6 $$

Now multiplying both sides of the above equation by $ x^2 + x $, we get

$$ 2x^2 + 2x + 1 = 6(x^2 + x) $$

$$ \Rightarrow 2x^2 + 2x + 1 = 6x^2 + 6x $$

$$ \Rightarrow 6x^2 – 2x^2 + 6x – 2x – 1 = 0 $$

$$ \Rightarrow 4x^2 + 4x – 1 = 0 \;\;\;\;\; (ii) $$

Equation (ii) is in the standard form of the quadratic equation but it is not a pure quadratic equation because b ≠ 0.

$ \pmb{ (iv). \;\;\;\;\; {x+4 \over x – 2} – {x – 2 \over x} + 4 = 0 } $

### Solution:

The given quadratic equation is

$$ {x + 4 \over x – 2} – {x – 2 \over x} + 4 = 0 \;\;\;\;\; (i) $$

Multiplying both sides of equation (i) by $ {(x – 2) (x)} $ (i.e. LCM of (x – 2) and x), we get

$$ (x – 2) (x){x + 4 \over x – 2} – (x – 2) (x){x – 2 \over x} + 4(x – 2) (x) = 0(x – 2) (x) $$

$$ \Rightarrow (x) (x + 4) – (x – 2) (x – 2) + 4(x^2 – 2x) = 0 $$

$$ \Rightarrow x^2 + 4x – (x^2 – 4x + 4) + 4x^2 – 8x = 0 $$

$$ \Rightarrow x^2 + 4x – x^2 + 4x – 4 + 4x^2 – 8x = 0 $$

$$ \Rightarrow 4x^2 – 4 = 0 $$

Taking 4 common in above equation, we have

$$ x^2 – 1 = 0 \;\;\;\;\; (ii) $$

Equation (ii) is in the standard form of the quadratic equation and it is also a pure quadratic equation because b = 0.