Question

Multiply each side of the equation by an appropriate power of ten to obtain integer coefficients. Then solve by factoring. $$0.3 n^{2}-2.2 n+8.4=0$$

Short answer

The solutions for the given quadratic equation are \(n = 6\) and \(n = \frac{14}{3}\).

Step-by-step solution

Removal of Decimal Coefficients

To remove the decimal coefficients from \(0.3 n^{2} - 2.2 n + 8.4 = 0\), the equation can be multiplied by a factor of 10. This will give a new equation \(3 n^{2} - 22 n + 84 = 0\).

Factoring the Equation

Now this quadratic equation \(3 n^{2} - 22 n + 84 = 0\) can be factored into two binomial equations by finding two numbers that multiply to give \(3 * 84 = 252\), and add to give -22. The two numbers fulfilling this condition are -14 and -18. Therefore, the factored form of the equation is \(3 n^{2} - 14 n - 18 n + 84 = 0\), that can be written as \(n(3n - 14) - 6(3n - 14) = 0\), giving the simplified factored form as \((n - 6)(3n - 14) = 0\).

Solving for n

\((n - 6)(3n - 14) = 0\) gives us two possible solutions: setting each binomial equation to zero and solving for 'n'. From \(n - 6 = 0\), we derive \(n = 6\). For \(3n - 14 = 0\), we get the solution \(n = \frac{14}{3}\).