Question

The value of $k$ for which the roots the real and equal of the following equation ${x^2} - 2(5 + 2k)x + 3(7 + 10k) = 0$ are $k = 2,\dfrac{1}{2}$
True or False?

Answer 1

Hint: A quadratic equation can be represented in a general form: $a{x^2} + bx + c = 0$ . The information about the nature of the roots is given by the value of the discriminant.
$D = {b^2} - 4ac$
If $D = 0$ then, the equation has real and equal roots.

Complete answer:
The given equation is ${x^2} - 2(5 + 2k)x + 3(7 + 10k) = 0$
Let’s start by writing down the coefficients of the equation:
$a = 1$
$b = - 2(5 + 2k)$
$c = 3(7 + 10k)$
Now using the formula of D, we get:
${[2(5 + 2k)]^2} - 4[3(7 + 10k)] = 0$
Now we get another quadratic equation in variable $k$
After simplifying the equation, we get:
$4[25 + 4{k^2} + 20k] - 4[21 + 30k] = 0$
After taking out 4 common from the equation, we get:
$25 + 4{k^2} + 20k - 21 - 30k = 0$
$4{k^2} - 10k + 4 = 0$
Using the ‘splitting the middle term’ method to solve the quadratic equations, we get:
$4{k^2} - 8k - 2k + 4 = 0$
$4k(k - 2) - 2(k - 2) = 0$
$(k - 2)(2k - 1) = 0$
The roots of the above equation are $k = 2,\dfrac{1}{2}$
Hence, the answer is True.

Note: A general quadratic equation can be written as: $a{x^2} + bx + c = 0$
The discriminant value is said to be: $D = {b^2} - 4ac$
$D > 0$ : Roots are real and distinct
$D = 0$ : Roots are real and equal
$D < 0$ : Roots are imaginary
Based on the above relations between the discriminant and the nature of the roots, similar questions like these can easily be solved. Hence remembering them will always come in handy. We can also check our answer by substituting the value of $k$in the main equation and try finding its roots.