Question

Use the quadratic formula to solve. Express your answer in simplest form.
10, x, squared, minus, 4, x, plus, 1, equals, 6, x, squared
10x
2
−4x+1=
6x
2

Answer 1

To use the quadratic formula to solve the equation \( 10x^2 - 4x + 1 = 6x^2 \), we first need to rearrange the equation into standard form \( ax^2 + bx + c = 0 \).

1. **Rearranging the equation**:

\( 10x^2 - 4x + 1 - 6x^2 = 0 \)

Combine like terms:

\( (10x^2 - 6x^2) - 4x + 1 = 0 \)

This simplifies to:

\( 4x^2 - 4x + 1 = 0 \)

2. **Identifying coefficients**:

Here, we have:
- \( a = 4 \)
- \( b = -4 \)
- \( c = 1 \)

3. **Using the quadratic formula**:

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Now, substitute the values of \( a \), \( b \), and \( c \):

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4}
\]

Simplifying this gives:

\[
x = \frac{4 \pm \sqrt{16 - 16}}{8}
\]

\[
x = \frac{4 \pm \sqrt{0}}{8}
\]

4. **Simplifying further**:

Since \( \sqrt{0} = 0 \), we have:

\[
x = \frac{4 \pm 0}{8}
\]

Thus:

\[
x = \frac{4}{8} = \frac{1}{2}
\]

5. **Conclusion**:

The solution to the equation \( 10x^2 - 4x + 1 = 6x^2 \) is

\[
x = \frac{1}{2}
\]

This is the simplest form of the solution.