Question

Determine the value of k for which the quadratic equation $4{x^2} - 3kx + 1 = 0$ has equal roots.
A. $ \pm \dfrac{2}{3}$
B. $ \pm \dfrac{4}{3}$
C. $ \pm 4$
D. $ \pm 6$

Answer 1

Hint:In the given question, we are required to solve for the value of k such that the equation $4{x^2} - 3kx + 1 = 0$ 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 quadratic equation $4{x^2} - 3kx + 1 = 0$.
Firstly, comparing the equation with standard form of a quadratic equation $a{x^2} + bx + c = 0$
Here,$a = 4$, $b = - 3k$ and$c = 1$.
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( { - 3m} \right)}^2} - 4\left( 4 \right)\left( 1 \right)} = 0$
$ \Rightarrow \sqrt {9{m^2} - 16} = 0$

Squaring both sides of equation, we get,
$ \Rightarrow 9{m^2} - 16 = 0$
Isolating m and finding its value, we get,
\[ \Rightarrow 9{m^2} = 16\]
Dividing both sides by nine,
\[ \Rightarrow {m^2} = \dfrac{{16}}{9}\]
Taking square root on both sides, we get,
\[ \Rightarrow m = \pm \sqrt {\dfrac{{16}}{9}} \]
\[ \therefore m = \pm \dfrac{4}{3}\]

Hence, option B is the correct answer.

Note:Quadratic equations are the polynomial equations with degree of the variable or unknown as two. We should also know the expression of the discriminant of a quadratic equation so as to solve the question. We can directly equate the discriminant to zero to find the value of $k$ as we are given that the roots are equal.