Question

For what values of k, the equation $9{x^2} - 24x + k = 0$ has equal roots? Find the roots.

Answer 1

Hint: In the given question, we are required to solve for the value of k such that the equation $9{x^2} - 24x + k = 0$ 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 the quadratic formula to find the condition for equal roots of a quadratic equation.

Complete step-by-step solution:
In the given question, we are provided with the equation $9{x^2} - 24x + k = 0$.
Now, comparing the equation with standard form of a quadratic equation $a{x^2} + bx + c = 0$
Here,$a = 9$, $b = - 24$ and$c = k$.
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( { - 24} \right)}^2} - 4\left( 9 \right)\left( k \right)} = 0$
$ \Rightarrow \sqrt {576 - 36k} = 0$
Factoring out $36$ from the expression and taking it out of the square root, we get,
$ \Rightarrow 6\sqrt {16 - k} = 0$
Now, dividing both the sides of equation by six and squaring both sides of the equation, we get,
$ \Rightarrow \sqrt {16 - k} = 0$
Squaring both sides of equation, we get,
$ \Rightarrow 16 - k = 0$
Now, shifting the terms in equation using method of transposition, we get,
$ \Rightarrow k = 16$
Hence, the value of k is $16$.

Note: We must know algebraic factorization and simplification rules in order to simplify the equation. One should know the expression for discriminant as ${b^2} - 4ac$ for a quadratic equation
$a{x^2} + bx + c = 0$. Care should be taken while doing the calculations in order to get to the final answer.