Solve for the roots in simplest form using the quadratic formula:
4, x, squared, plus, 5, equals, minus, 12, x
4x 
2
 +5=
−12x

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.

Solve for the roots in simplest form using the quadratic formula: 4x^2 + 5 = -12x

This page explains how to solve the quadratic equation 4x^2 + 12x + 5 using the quadratic formula, providing step-by-step guidance on finding the roots. Learn key algebraic concepts like the discriminant and apply the quadratic formula theorem. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation