Question

Use the quadratic formula to find the solutions to the equation.
x2-3x+1=0
5土V10
3
3土v5
2
3士v15
2
2+\24
2

Answer 1

To solve the equation \( x^2 - 3x + 1 = 0 \) using the quadratic formula, we will follow these steps:

The quadratic formula is given by:

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

For the equation \( x^2 - 3x + 1 = 0 \):

- \( a = 1 \)
- \( b = -3 \)
- \( c = 1 \)

Now, we can substitute \( a \), \( b \), and \( c \) into the formula:

1. Calculate the discriminant (\( b^2 - 4ac \)):
\[
b^2 - 4ac = (-3)^2 - 4 \cdot 1 \cdot 1 = 9 - 4 = 5
\]

2. Now plug the values into the quadratic formula:
\[
x = \frac{{-(-3) \pm \sqrt{5}}}{{2 \cdot 1}} = \frac{{3 \pm \sqrt{5}}}{2}
\]

Thus, the solutions are:

\[
x = \frac{{3 + \sqrt{5}}}{2} \quad \text{and} \quad x = \frac{{3 - \sqrt{5}}}{2}
\]

Among the options provided, neither \( 5\sqrt{10}/3 \), \( 3 \sqrt{5}/2 \), \( 3 \sqrt{15}/2 \), nor \( 2+\sqrt{24}/2 \) directly relate to the computed solutions.

Therefore, the solutions to the equation \( x^2 - 3x + 1 = 0 \) are:

\[
x = \frac{{3 + \sqrt{5}}}{2} \quad \text{and} \quad x = \frac{{3 - \sqrt{5}}}{2}
\]