Question

Solve the quadratic equation. Show all of your steps. x2+3x−5=0 copy and paste format

Answer 1

To solve the quadratic equation \( x^2 + 3x - 5 = 0 \), we can use the quadratic formula, which is given by:

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

where \( a \), \( b \), and \( c \) are the coefficients from the quadratic equation in the standard form \( ax^2 + bx + c = 0 \).

For the equation \( x^2 + 3x - 5 = 0 \):

- \( a = 1 \)
- \( b = 3 \)
- \( c = -5 \)

Now, let's substitute the values of \( a \), \( b \), and \( c \) into the quadratic formula:

1. Calculate the discriminant \( b^2 - 4ac \):

\[
b^2 = 3^2 = 9
\]
\[
4ac = 4 \cdot 1 \cdot (-5) = -20
\]
\[
b^2 - 4ac = 9 - (-20) = 9 + 20 = 29
\]

2. Now substituting back into the quadratic formula:

\[
x = \frac{{-3 \pm \sqrt{29}}}{2 \cdot 1}
\]
\[
x = \frac{{-3 \pm \sqrt{29}}}{2}
\]

3. So the solutions are:

\[
x_1 = \frac{{-3 + \sqrt{29}}}{2}
\]
\[
x_2 = \frac{{-3 - \sqrt{29}}}{2}
\]

Thus, the solutions to the equation \( x^2 + 3x - 5 = 0 \) are:

\[
x_1 = \frac{{-3 + \sqrt{29}}}{2}, \quad x_2 = \frac{{-3 - \sqrt{29}}}{2}
\]