Certainly! Let's go through the quadratic formula and the calculation of the discriminant step-by-step.
A quadratic equation is generally expressed in the form:
[tex]\[ ax^2 + bx + c = \0 \][/tex]
The quadratic formula for solving this equation for \( x \) is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \][/tex]
To use this formula, we first need to calculate the discriminant, which is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
The discriminant helps us determine the nature of the roots of the quadratic equation:
1. If \(\Delta > 0\), the equation has two distinct real roots.
2. If \(\Delta = 0\), the equation has exactly one real root (a repeated root).
3. If \(\Delta < 0\), the equation has two complex roots.
Let's assume \( a = 1 \), \( b = 1 \), and \( c = 1 \). These are the coefficients of the quadratic equation.
Step-by-Step Solution:
1. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (1)^2 - 4 \cdot 1 \cdot 1 \][/tex]
[tex]\[ \Delta = 1 - 4 \][/tex]
[tex]\[ \Delta = -3 \][/tex]
2. Determine the type of roots based on the discriminant (\(\Delta\)):
Since \(\Delta = -3\), which is less than 0, the quadratic equation will have two complex roots.
3. Use the quadratic formula to find the roots:
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since \(\Delta\) is negative, we will end up with an imaginary part in our solution.
Plugging in the values, we get:
[tex]\[ x = \frac{-1 \pm \sqrt{-3}}{2 \cdot 1} \][/tex]
[tex]\[ x = \frac{-1 \pm i\sqrt{3}}{2} \][/tex]
4. Express the roots:
The two roots are:
[tex]\[ x_1 = \frac{-1 + i\sqrt{3}}{2} \][/tex]
[tex]\[ x_2 = \frac{-1 - i\sqrt{3}}{2} \][/tex]
So, the solutions to the quadratic equation \( x^2 + x + 1 = 0 \) are:
[tex]\[ x_1 = \frac{-1 + i\sqrt{3}}{2} \][/tex]
[tex]\[ x_2 = \frac{-1 - i\sqrt{3}}{2} \][/tex]
And the discriminant is [tex]\(\Delta = -3\)[/tex]. These are the detailed steps and the final solutions for the quadratic equation using the given coefficients.
