Solve 312 = n[- 3n + 57] | Equatix - Math Solver

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Example

3(r+2s)=2t-4

a⋅(b-2)=3b

3(r+2s)=2t-4

a⋅(b-2)=3b

Question

$$312=n[-3n+57]$$

Solve for n

$n=\frac{5+\sqrt{3719}i}{6}\approx 0.833333333+10.163934059i$
$n=\frac{-\sqrt{3719}i+5}{6}\approx 0.833333333-10.163934059i$

Steps Using the Quadratic Formula

Steps for Completing the Square

Steps Using the Quadratic Formula

Use the distributive property to multiply $n$ by $-3n+5$.

$$312=-3n^{2}+5n$$

Swap sides so that all variable terms are on the left hand side.

$$-3n^{2}+5n=312$$

Subtract $312$ from both sides.

$$-3n^{2}+5n-312=0$$

This equation is in standard form: $ax^{2}+bx+c=0$. Substitute $-3$ for $a$, $5$ for $b$, and $-312$ for $c$ in the quadratic formula, $\frac{-b±\sqrt{b^{2}-4ac}}{2a}$.

$$n=\frac{-5±\sqrt{5^{2}-4\left(-3\right)\left(-312\right)}}{2\left(-3\right)}$$

Square $5$.

$$n=\frac{-5±\sqrt{25-4\left(-3\right)\left(-312\right)}}{2\left(-3\right)}$$

Multiply $-4$ times $-3$.

$$n=\frac{-5±\sqrt{25+12\left(-312\right)}}{2\left(-3\right)}$$

Multiply $12$ times $-312$.

$$n=\frac{-5±\sqrt{25-3744}}{2\left(-3\right)}$$

Add $25$ to $-3744$.

$$n=\frac{-5±\sqrt{-3719}}{2\left(-3\right)}$$

Take the square root of $-3719$.

$$n=\frac{-5±\sqrt{3719}i}{2\left(-3\right)}$$

Multiply $2$ times $-3$.

$$n=\frac{-5±\sqrt{3719}i}{-6}$$

Now solve the equation $n=\frac{-5±\sqrt{3719}i}{-6}$ when $±$ is plus. Add $-5$ to $i\sqrt{3719}$.

$$n=\frac{-5+\sqrt{3719}i}{-6}$$

Divide $-5+i\sqrt{3719}$ by $-6$.

$$n=\frac{-\sqrt{3719}i+5}{6}$$

Now solve the equation $n=\frac{-5±\sqrt{3719}i}{-6}$ when $±$ is minus. Subtract $i\sqrt{3719}$ from $-5$.

$$n=\frac{-\sqrt{3719}i-5}{-6}$$

Divide $-5-i\sqrt{3719}$ by $-6$.

$$n=\frac{5+\sqrt{3719}i}{6}$$

The equation is now solved.

$$n=\frac{-\sqrt{3719}i+5}{6}$$ $$n=\frac{5+\sqrt{3719}i}{6}$$

Steps for Completing the Square

Use the distributive property to multiply $n$ by $-3n+5$.

$$312=-3n^{2}+5n$$

Swap sides so that all variable terms are on the left hand side.

$$-3n^{2}+5n=312$$

Divide both sides by $-3$.

$$\frac{-3n^{2}+5n}{-3}=\frac{312}{-3}$$

Dividing by $-3$ undoes the multiplication by $-3$.

$$n^{2}+\frac{5}{-3}n=\frac{312}{-3}$$

Divide $5$ by $-3$.

$$n^{2}-\frac{5}{3}n=\frac{312}{-3}$$

Divide $312$ by $-3$.

$$n^{2}-\frac{5}{3}n=-104$$

Divide $-\frac{5}{3}$, the coefficient of the $x$ term, by $2$ to get $-\frac{5}{6}$. Then add the square of $-\frac{5}{6}$ to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{5}{3}n+\left(-\frac{5}{6}\right)^{2}=-104+\left(-\frac{5}{6}\right)^{2}$$

Square $-\frac{5}{6}$ by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{5}{3}n+\frac{25}{36}=-104+\frac{25}{36}$$

Add $-104$ to $\frac{25}{36}$.

$$n^{2}-\frac{5}{3}n+\frac{25}{36}=-\frac{3719}{36}$$

Factor $n^{2}-\frac{5}{3}n+\frac{25}{36}$. In general, when $x^{2}+bx+c$ is a perfect square, it can always be factored as $\left(x+\frac{b}{2}\right)^{2}$.

$$\left(n-\frac{5}{6}\right)^{2}=-\frac{3719}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{5}{6}\right)^{2}}=\sqrt{-\frac{3719}{36}}$$

Simplify.

$$n-\frac{5}{6}=\frac{\sqrt{3719}i}{6}$$ $$n-\frac{5}{6}=-\frac{\sqrt{3719}i}{6}$$

Add $\frac{5}{6}$ to both sides of the equation.

$$n=\frac{5+\sqrt{3719}i}{6}$$ $$n=\frac{-\sqrt{3719}i+5}{6}$$