To solve the quadratic equation [tex]\( x^2 + 4x - 3 = 0 \)[/tex], we will use the quadratic formula. The quadratic formula states that for any quadratic equation of the form [tex]\(ax^2 + bx + c = 0\)[/tex], the solutions can be found using:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = 4 \)[/tex]
- [tex]\( c = -3 \)[/tex]
### Step-by-Step Solution
1. Calculate the Discriminant:
The discriminant ([tex]\(\Delta\)[/tex]) is calculated by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the given coefficients:
[tex]\[ \Delta = 4^2 - 4 \cdot 1 \cdot (-3) \][/tex]
[tex]\[ \Delta = 16 + 12 \][/tex]
[tex]\[ \Delta = 28 \][/tex]
2. Calculate the Square Root of the Discriminant:
We need to find [tex]\(\sqrt{\Delta}\)[/tex]:
[tex]\[ \sqrt{28} \approx 5.291502622129181 \][/tex]
3. Calculate the Two Solutions:
Using the quadratic formula, we find the two solutions ([tex]\(x_1\)[/tex] and [tex]\(x_2\)[/tex]):
[tex]\[
x_1 = \frac{-b + \sqrt{\Delta}}{2a}
\][/tex]
[tex]\[
x_2 = \frac{-b - \sqrt{\Delta}}{2a}
\][/tex]
Substituting the values:
[tex]\[ x_1 = \frac{-4 + 5.291502622129181}{2 \cdot 1} \][/tex]
[tex]\[ x_1 = \frac{1.291502622129181}{2} \][/tex]
[tex]\[ x_1 \approx 0.6457513110645907 \][/tex]
[tex]\[ x_2 = \frac{-4 - 5.291502622129181}{2 \cdot 1} \][/tex]
[tex]\[ x_2 = \frac{-9.291502622129181}{2} \][/tex]
[tex]\[ x_2 \approx -4.645751311064591 \][/tex]
### Summary of Results:
- The discriminant is: [tex]\( \Delta = 28 \)[/tex]
- The two solutions are approximately:
- [tex]\( x_1 \approx 0.6457513110645907 \)[/tex]
- [tex]\( x_2 \approx -4.645751311064591 \)[/tex]
These are the roots of the quadratic equation [tex]\( x^2 + 4x - 3 = 0 \)[/tex].
