Question

question
use the quadratic formula to solve. express your answer in simplest form.
25n² - 15n + 4 = 5n

Explanation:

Step1: Rewrite the equation in standard form \(ax^2 + bx + c = 0\)

First, we subtract \(5n\) from both sides of the equation \(25n^2 - 15n + 4 = 5n\) to get it in standard quadratic form.
\[

$$\begin{align*} 25n^2 - 15n + 4 - 5n&= 5n - 5n\\ 25n^2 - 20n + 4&= 0 \end{align*}$$

\]
Here, \(a = 25\), \(b = - 20\), and \(c = 4\).

Step2: Recall the quadratic formula

The quadratic formula for a quadratic equation \(ax^2+bx + c = 0\) is given by \(n=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\)

Step3: Calculate the discriminant \(\Delta=b^{2}-4ac\)

Substitute \(a = 25\), \(b=-20\) and \(c = 4\) into the discriminant formula.
\[

$$\begin{align*} \Delta&=(-20)^{2}-4\times25\times4\\ &=400 - 400\\ &= 0 \end{align*}$$

\]

Step4: Substitute the values of \(a\), \(b\) and \(\Delta\) into the quadratic formula

Since \(\Delta = 0\), we have:
\[

$$\begin{align*} n&=\frac{-(-20)\pm\sqrt{0}}{2\times25}\\ &=\frac{20\pm0}{50}\\ &=\frac{20}{50} \end{align*}$$

\]

Step5: Simplify the fraction

Simplify \(\frac{20}{50}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 10.
\[
\frac{20\div10}{50\div10}=\frac{2}{5}
\]

Answer:

\(n = \frac{2}{5}\) (Since the discriminant is zero, there is a repeated root)