If \(4x^{2} + 5kx + 10\) is a perfect square, find the value of k.
Question 1 Report
If \(4x^{2} + 5kx + 10\) is a perfect square, find the value of k.
Answer Details
We know that a perfect square trinomial has the form of \((ax + b)^2\), where a and b are constants.
So, if we have a trinomial of the form \(4x^2 + 5kx + 10\), we can write it as \((2x + c)^2\), where c is another constant.
Expanding the square, we get:
\begin{align*}
(2x + c)^2 &= 4x^2 + 4cx + c^2 \\
&= 4x^2 + (4c)x + c^2 \\
\end{align*}
Comparing the coefficients of the two trinomials, we get:
\begin{align*}
4 &= 4 \\
5k &= 4c \\
10 &= c^2
\end{align*}
Dividing the second equation by 4, we get:
\begin{align*}
\frac{5k}{4} &= c
\end{align*}
Since c = \( \sqrt{10} \), we get:
\begin{align*}
\frac{5k}{4} &= \sqrt{10} \\
\Rightarrow k &= \frac{4\sqrt{10}}{5}
\end{align*}
So the value of k is \( \frac{4\sqrt{10}}{5} \).
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