factor by completing the square the equation (4x+1)(x+2)=-2

To factor the equation \((4x + 1)(x + 2) = -2\) by completing the square, we will first expand the left-hand side, rearrange it into a standard quadratic form, and then complete the square.

### Step 1: Expand and rearrange the equation

Let's expand the left side of the equation:

\[
(4x + 1)(x + 2) = 4x^2 + 8x + x + 2 = 4x^2 + 9x + 2
\]

So, the equation becomes:

\[
4x^2 + 9x + 2 = -2
\]

Now, add 2 to both sides to set the equation to zero:

\[
4x^2 + 9x + 4 = 0
\]

### Step 2: Complete the square

To complete the square, we want to rewrite the quadratic expression \(4x^2 + 9x + 4\) in the form \(a(x - h)^2 + k = 0\).

First, divide the entire equation by 4 to simplify:

\[
x^2 + \frac{9}{4}x + 1 = 0
\]

Next, to complete the square, we need to take half of the coefficient of \(x\), square it, and add and subtract it inside the equation.

The coefficient of \(x\) is \(\frac{9}{4}\). Half of this is \(\frac{9}{8}\), and squaring it gives:

\[
\left(\frac{9}{8}\right)^2 = \frac{81}{64}
\]

Now, add and subtract this value inside the equation:

\[
x^2 + \frac{9}{4}x + \frac{81}{64} - \frac{81}{64} + 1 = 0
\]

Rewrite the expression:

\[
\left(x + \frac{9}{8}\right)^2 - \frac{81}{64} + 1 = 0
\]

Combine the constants:

\[
\left(x + \frac{9}{8}\right)^2 - \frac{81}{64} + \frac{64}{64} = 0
\]

\[
\left(x + \frac{9}{8}\right)^2 - \frac{17}{64} = 0
\]

### Step 3: Solve for \(x\)

Now, isolate the square term:

\[
\left(x + \frac{9}{8}\right)^2 = \frac{17}{64}
\]

Take the square root of both sides:

\[
x + \frac{9}{8} = \pm \sqrt{\frac{17}{64}}
\]

Since \(\sqrt{\frac{17}{64}} = \frac{\sqrt{17}}{8}\), we have:

\[
x + \frac{9}{8} = \pm \frac{\sqrt{17}}{8}
\]

Solve for \(x\):

\[
x = -\frac{9}{8} \pm \frac{\sqrt{17}}{8}
\]

This gives us the two solutions:

\[
x = \frac{-9 + \sqrt{17}}{8} \quad \text{and} \quad x = \frac{-9 - \sqrt{17}}{8}
\]

### Final Answer

The factored form of the equation is:

\[
4 \left(x - \frac{-9 + \sqrt{17}}{8}\right) \left(x - \frac{-9 - \sqrt{17}}{8}\right) = 0
\]

Or equivalently:

\[
4 \left(x + \frac{9 - \sqrt{17}}{8}\right) \left(x + \frac{9 + \sqrt{17}}{8}\right) = 0
\]

Would you like more details or have any questions?

### Related Questions:

1. What is the method of completing the square?
2. How do you factor a quadratic equation by different methods?
3. How can you solve quadratic equations using the quadratic formula?
4. What are the properties of square roots when solving equations?
5. How do you convert between different forms of quadratic expressions?

**Tip:** Always double-check the arithmetic when completing the square to avoid errors in coefficients or constants.

Learn how to factor the quadratic equation (4x+1)(x+2)=-2 using the method of completing the square. This detailed step-by-step solution explains how to expand, rearrange, and complete the square, involving key algebraic concepts and quadratic formulas. Ideal for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation