Questions: Find the solution of the following using the quadratic formula. x^2+2 x-5=0 x=-1 pm 2 i x=-1 pm sqrt6 x=1 pm sqrt6 x=-1 pm 2 sqrt6

Solution Steps

To solve the quadratic equation \(x^2 + 2x - 5 = 0\) using the quadratic formula, we need to identify the coefficients \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\). Then, we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions.

1. Identify the coefficients \(a\), \(b\), and \(c\) from the equation \(x^2 + 2x - 5 = 0\).
2. Substitute these coefficients into the quadratic formula.
3. Calculate the discriminant \(b^2 - 4ac\).
4. Compute the two possible values for \(x\) using the quadratic formula.

The given quadratic equation is

\[ x^2 + 2x - 5 = 0 \]

From this equation, we identify the coefficients as follows:

• \(a = 1\)
• \(b = 2\)
• \(c = -5\)

We calculate the discriminant using the formula

\[ D = b^2 - 4ac \]

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

\[ D = 2^2 - 4 \cdot 1 \cdot (-5) = 4 + 20 = 24 \]

Using the quadratic formula

\[ x = \frac{-b \pm \sqrt{D}}{2a} \]

we substitute \(b = 2\) and \(D = 24\):

\[ x = \frac{-2 \pm \sqrt{24}}{2 \cdot 1} \]

Calculating the square root and simplifying:

\[ \sqrt{24} = 2\sqrt{6} \]

Thus, the solutions become:

\[ x = \frac{-2 \pm 2\sqrt{6}}{2} = -1 \pm \sqrt{6} \]

This gives us two solutions:

\[ x_1 = -1 + \sqrt{6} \quad \text{and} \quad x_2 = -1 - \sqrt{6} \]

Final Answer