To solve for the roots using the quadratic formula, we first need to rewrite the given equation in the standard form $ax^2 + bx + c = 0$.

Step 1: Rewrite the equation

The given equation is: $4x^2 + 5 = -12x$

Move all terms to one side of the equation to get: $4x^2 + 12x + 5 = 0$

Now, we have it in the form $ax^2 + bx + c = 0$, where:

• $a = 4$
• $b = 12$
• $c = 5$

Step 2: Apply the quadratic formula

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

Step 3: Substitute $a$, $b$, and $c$

Plug in the values: $x = \frac{-12 \pm \sqrt{12^2 - 4 \cdot 4 \cdot 5}}{2 \cdot 4}$

Step 4: Simplify under the square root

Calculate $b^2 - 4ac$: $12^2 - 4 \cdot 4 \cdot 5 = 144 - 80 = 64$

Step 5: Simplify the equation

The equation becomes: $x = \frac{-12 \pm \sqrt{64}}{8}$ Since $\sqrt{64} = 8$, we get: $x = \frac{-12 \pm 8}{8}$

Step 6: Find the two solutions

1. $x = \frac{-12 + 8}{8} = \frac{-4}{8} = -\frac{1}{2}$
2. $x = \frac{-12 - 8}{8} = \frac{-20}{8} = -\frac{5}{2}$

Final Answer

The roots of the quadratic equation are: $x = -\frac{1}{2} \quad \text{and} \quad x = -\frac{5}{2}$

Would you like more details or have any questions?

Here are five related questions to consider:

1. How does the discriminant $b^2 - 4ac$ determine the nature of the roots?
2. What are the steps to complete the square for solving quadratic equations?
3. How can the quadratic formula be derived from the standard form of a quadratic equation?
4. What are some real-world applications of quadratic equations?
5. How can graphing a quadratic function help in understanding its roots?

Tip: When solving a quadratic equation, always simplify the equation to the standard form $ax^2 + bx + c = 0$ before applying any solution method.