For what value of A, does (k – 12)x² + 2 (k – 12)x + 2 = 0 have equal roots?

Here, a = k – 12, b = 2 (k – 12), c = 2
∴ D = b² – 4ac = [2(k– 12)]²
– 4(k – 12) x 2
= 4 (k – 12)² – 8 (k– 12)
Roots are equal, if D = 0
⇒ 4 (k – 12) 2 – 8 (k – 12) = 0
⇒ 4(k – 12)(k – 12 – 2) = 0
⇒ (k – 12) (k – 14) = 0
⇒ k = 12 or 14
But k = 12 does not satisfy the eqn.
k = 14 Ans.

Let the number of toys produced on that day = x
And, cost of production = (55 – x)
According to the given question,
x (55 – x) = 750
⇒ 55x – x² – 750 = 0
x² – 55x + 750 = 0
⇒ x² – 30x – 25x + 750 = 0
⇒ x(x – 30) – 25(x – 30) = 0
⇒ (x – 30) (x – 25) = 0
⇒ x – 30 = 0 or x – 25 = 0
⇒ x = 30 or x = 25.
Hence, the number of toys produced on that day are 30 or 25.

∵ (–5) is a root of the quadratic equation
∴ 2x² + px – 15 = 0
2(–5)² + p. (–5) – 15 = 0
⇒ 50 – 5p – 15 = 0
⇒ –5p + 35= 0
⇒ –5p = –35
⇒ P = 7
Now. the quadratic equation 7(x² + x) + k = 0
i.e. 7x² + 7x + k = 0 has equal roots.
∴      Its discriminant  
            
              

Hence,   

Let the roots of the quadratic equation be α and β.
We have,
2x² + px – 4 = 0, then
Product of the roots = 
       
But        

        
Sum of the roots =