Question

If the following quadratic equation has two equal and real roots then find the value of $k$:
\[k{{x}^{2}}-24x+16=0\]

Answer 1

Hint: The given quadratic equation has two equal and real roots. From this information, we can deduce that we have to use the properties of the discriminant to solve this question. If the discriminant is $0$, then the quadratic equation has two equal and real roots. Using this fact, we will obtain a linear equation with $k$ as the variable.

Complete step-by-step solution:
The general quadratic equation is $a{{x}^{2}}+bx+c=0$, where $a\ne 0$. The discriminant of this general quadratic equation is given by
\[D={{b}^{2}}-4ac\]
Now, we know that the discriminant has the following property:
If the discriminant $D=0$, then the quadratic equation has two equal and real roots.
The equation we are given is \[k{{x}^{2}}-24x+16=0\]. We know that this equation has two equal and real roots. Hence, the discriminant of the given equation is $0$. Comparing the given equation with the general quadratic equation, we have the following values:
\[\begin{align}
  & a=k\ne 0 \\
 & b=-24 \\
 & c=16 \\
\end{align}\]
Now, we will use the fact that the discriminant of the given equation is $0$. We will substitute the values of $a$, $b$ and $c$ in the equation ${{b}^{2}}-4ac=0$. So we will get the following equation,
\[{{(-24)}^{2}}-4\cdot k\cdot 16=0\]
Simplifying this equation we get,
\[576-64\cdot k=0\]
Rearranging the terms, we get the value of $k$ as follows,
\[64\cdot k=576\]
Therefore, $k=\dfrac{576}{64}$ which implies that $k=9$.
Hence, the value of $k$ is $9$.

Note: The properties of the discriminant are that: (1) if $D >0$, then the quadratic equation has two distinct, real roots; (2) if $D< 0$, then the quadratic equation has roots that are not real; and (3) if $D=0$, then the two roots are equal and real. Substituting the correct values without missing the signs for $a$, $b$ and $c$ is important. If the signs are not taken into account correctly, the value of the discriminant will not give us the correct interpretation of the roots of the quadratic equation.