Question

If the equation $ {x^2} - m\left( {2x - 8} \right) - 15 = 0 $ has equal roots then m equals:
(A) $ 3, - 5 $
(B) $ - 3,5 $
(C) $ 3,5 $
(D) $ - 3, - 5 $

Answer 1

Hint: In the given question, we are required to solve for the value of m such that the equation $ {x^2} - m\left( {2x - 8} \right) - 15 = 0 $ has equal roots. We will first compare the given equation with the standard form of a quadratic equation $ a{x^2} + bx + c = 0 $ and then apply a quadratic formula to find the condition for equal roots of a quadratic equation.

Complete step-by-step answer:
In the given question, we are provided with the equation $ {x^2} - m\left( {2x - 8} \right) - 15 = 0 $ .
Firstly, simplifying the equation by opening the brackets, we get,
 $ \Rightarrow {x^2} - 2mx + 8m - 15 = 0 $
Now, comparing the equation with standard form of a quadratic equation $ a{x^2} + bx + c = 0 $
Here, $ a = 1 $ , $ b = - 2m $ and $ c = 8m - 15 $ .
Now, using the quadratic formula, we get the roots of the equation as:
 $ x = \dfrac{{( - b) \pm \sqrt {{b^2} - 4ac} }}{{2a}} $
If both the roots of a quadratic equation are equal, then, we get,
 $ {x_1} = {x_2} $
 $ \Rightarrow \dfrac{{( - b) + \sqrt {{b^2} - 4ac} }}{{2a}} = \dfrac{{( - b) - \sqrt {{b^2} - 4ac} }}{{2a}} $
Cross multiplying the terms of equation and simplifying further, we get,
 $ \Rightarrow \sqrt {{b^2} - 4ac} = - \sqrt {{b^2} - 4ac} $
Shifting all the terms to left side and dividing both sides of equation by two, we get,
 $ \Rightarrow \sqrt {{b^2} - 4ac} = 0 $
Now, we can substitute the values of a, b and c in the expression. So, we get,
 $ \Rightarrow \sqrt {{{\left( { - 2m} \right)}^2} - 4\left( 1 \right)\left( {8m - 15} \right)} = 0 $
 $ \Rightarrow \sqrt {4{m^2} - 32m + 60} = 0 $
Factoring out $ 4 $ from the expression and taking it out of the square root, we get,
 $ \Rightarrow 2\sqrt {{m^2} - 8m + 15} = 0 $
Now, dividing both the sides of equation by two and squaring both sides of the equation, we get,
 $ \Rightarrow {m^2} - 8m + 15 = 0 $
Now, splitting up the middle term to factorize the equation, we get,
 $ \Rightarrow {m^2} - 3m - 5m + 15 = 0 $
Taking m common from first two terms and $ - 5 $ from last two terms, we get,
 $ \Rightarrow m\left( {m - 3} \right) - 5\left( {m - 3} \right) = 0 $
 $ \Rightarrow \left( {m - 5} \right)\left( {m - 3} \right) = 0 $
Since the product of two terms is equal to zero. So, either of the two terms have to be zero. So, we get,
Either $ m - 5 = 0 $ or $ m - 3 = 0 $
So, we get, either $ m = 5 $ or $ m = 3 $
Hence, the values of m are: $ 3 $ and $ 5 $ .
Hence, the option (C) is the correct answer.
So, the correct answer is “Option C”.

Note: Quadratic equations are the polynomial equations with degree of the variable or unknown as 2. We should also know the expression of the discriminant of a quadratic equation so as to solve the question. We should take care of the calculations to be sure of the final answer.