Questions: Solve the equation by applying the quadratic formula. 5/12 x^2 - 1/2 x + 1/4 = 0

Solution Steps

To solve the quadratic equation using the quadratic formula, we need to identify the coefficients $a$, $b$, and $c$ from the equation $ax^2 + bx + c = 0$. Then, we apply the quadratic formula: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ to find the roots of the equation.

The given quadratic equation is: $$\frac{5}{12} x^{2} - \frac{1}{2} x + \frac{1}{4} = 0$$

We need to identify the coefficients $a$, $b$, and $c$ from the standard form of a quadratic equation $ax^2 + bx + c = 0$.

Here: $$a = \frac{5}{12}, \quad b = -\frac{1}{2}, \quad c = \frac{1}{4}$$

The quadratic formula is given by: $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The discriminant $\Delta$ is: $$\Delta = b^2 - 4ac$$

Substitute the values of $a$, $b$, and $c$: $$\Delta = \left(-\frac{1}{2}\right)^2 - 4 \left(\frac{5}{12}\right) \left(\frac{1}{4}\right)$$

Calculate each term: $$\left(-\frac{1}{2}\right)^2 = \frac{1}{4}$$ $$4 \left(\frac{5}{12}\right) \left(\frac{1}{4}\right) = \frac{5}{12} \cdot \frac{1}{4} \cdot 4 = \frac{5}{12}$$

So, $$\Delta = \frac{1}{4} - \frac{5}{12}$$

To subtract these fractions, find a common denominator: $$\frac{1}{4} = \frac{3}{12}$$

Thus, $$\Delta = \frac{3}{12} - \frac{5}{12} = -\frac{2}{12} = -\frac{1}{6}$$

Since the discriminant $\Delta$ is negative, the quadratic equation has two complex roots.

Substitute $a$, $b$, and $\Delta$ into the quadratic formula: $$x = \frac{-b \pm \sqrt{\Delta}}{2a}$$

Since $\Delta = -\frac{1}{6}$, we have: $$\sqrt{\Delta} = \sqrt{-\frac{1}{6}} = i \sqrt{\frac{1}{6}} = \frac{i}{\sqrt{6}} = \frac{i \sqrt{6}}{6}$$

Now, substitute $b$ and $a$: $$x = \frac{\frac{1}{2} \pm \frac{i \sqrt{6}}{6}}{\frac{10}{12}} = \frac{\frac{1}{2} \pm \frac{i \sqrt{6}}{6}}{\frac{5}{6}}$$

Simplify the fraction: $$x = \frac{1}{2} \cdot \frac{6}{5} \pm \frac{i \sqrt{6}}{6} \cdot \frac{6}{5} = \frac{3}{5} \pm \frac{i \sqrt{6}}{5}$$

Final Answer