Question

Solve the following quadratic equation using the perfect square trinomial pattern: 14x2+2x+4=0 .(1 point)

Answer 1

To solve the quadratic equation \( 14x^2 + 2x + 4 = 0 \) using the perfect square trinomial pattern, we can start by factoring out the greatest common factor (GCF) from the equation.

1. **Factor out the GCF (if applicable)**:
The GCF of the coefficients \( 14 \), \( 2 \), and \( 4 \) is \( 2 \). So, we factor out \( 2 \):

\[
2(7x^2 + x + 2) = 0
\]

Now we will solve the equation inside the parentheses:

\[
7x^2 + x + 2 = 0
\]

2. **Rearrangement and completing the square**:
To complete the square for \( 7x^2 + x + 2 \), we can rearrange the quadratic part:

First, divide the entire equation by \( 7 \) (this will help with completing the square):

\[
x^2 + \frac{1}{7}x + \frac{2}{7} = 0
\]

Next, we will move \( \frac{2}{7} \) to the other side:

\[
x^2 + \frac{1}{7}x = -\frac{2}{7}
\]

3. **Completing the square**:
To complete the square, take half of the coefficient of \( x \) (which is \( \frac{1}{7} \)), square it, and add it to both sides.

Half of \( \frac{1}{7} \) is \( \frac{1}{14} \). Squaring it gives:

\[
\left(\frac{1}{14}\right)^2 = \frac{1}{196}
\]

Now add \( \frac{1}{196} \) to both sides:

\[
x^2 + \frac{1}{7}x + \frac{1}{196} = -\frac{2}{7} + \frac{1}{196}
\]

To combine the fractions on the right side, we convert \( -\frac{2}{7} \) to have a denominator of \( 196 \):

\[
-\frac{2}{7} = -\frac{56}{196}
\]

Now combine:

\[
-\frac{56}{196} + \frac{1}{196} = -\frac{55}{196}
\]

So our equation now looks like this:

\[
\left(x + \frac{1}{14}\right)^2 = -\frac{55}{196}
\]

4. **Taking the square root**:
Now we proceed by taking the square root of both sides:

\[
x + \frac{1}{14} = \pm \sqrt{-\frac{55}{196}}
\]

The square root of a negative number involves imaginary numbers:

\[
x + \frac{1}{14} = \pm \frac{\sqrt{55}}{14}i
\]

5. **Solve for \( x \)**:
Finally, isolate \( x \):

\[
x = -\frac{1}{14} \pm \frac{\sqrt{55}}{14} i
\]

Thus, the solutions to the equation \( 14x^2 + 2x + 4 = 0 \) are:

\[
x = -\frac{1}{14} + \frac{\sqrt{55}}{14} i \quad \text{and} \quad x = -\frac{1}{14} - \frac{\sqrt{55}}{14} i
\]