Finding Roots of the Quadratic Equation 5m² + 18m + 16 = 0

The problem asks us to find the roots of the quadratic equation $5m^2 + 18m + 16 = 0$. A root of an equation is a value of the variable (in this case, $m$) that makes the equation true.

Using the Quadratic Formula

We can solve this quadratic equation using the quadratic formula, which is used to find the solutions for equations of the form $am^2 + bm + c = 0$. The formula is:

$$m = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

In our equation, $5m^2 + 18m + 16 = 0$, we have:

• $a = 5$
• $b = 18$
• $c = 16$

Step 1: Calculate the Discriminant

First, let's find the discriminant ($\Delta$), which is the part under the square root in the quadratic formula ($b^2 - 4ac$). This helps determine the nature of the roots.

$$\Delta = b^2 - 4ac$$

Substitute the values of $a$, $b$, and $c$:

$$\Delta = (18)^2 - 4(5)(16)$$

$$\Delta = 324 - 320$$

$$\Delta = 4$$

Step 2: Apply the Quadratic Formula

Now, substitute the values of $a$, $b$, and the discriminant ($\Delta$) back into the quadratic formula:

$$m = \frac{-18 \pm \sqrt{4}}{2(5)}$$

$$m = \frac{-18 \pm 2}{10}$$

Step 3: Calculate the Two Roots

We get two possible values for $m$ by using the plus and minus signs:

Root 1 (using '+'):

$$m_1 = \frac{-18 + 2}{10}$$

$$m_1 = \frac{-16}{10}$$

Simplify the fraction:

$$m_1 = -\frac{8}{5}$$

Root 2 (using '-'):

$$m_2 = \frac{-18 - 2}{10}$$

$$m_2 = \frac{-20}{10}$$

Simplify the fraction:

$$m_2 = -2$$

Conclusion

The roots of the quadratic equation $5m^2 + 18m + 16 = 0$ are $-\frac{8}{5}$ and $-2$. This corresponds to option 4.