To solve for the roots of the quadratic equation [tex]\(x^2 + 3x - 6 = 0\)[/tex] using the Quadratic Formula, we start by identifying the coefficients in the given equation.
The standard form of a quadratic equation is:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
In our equation, [tex]\(x^2 + 3x - 6 = 0\)[/tex], the coefficients are as follows:
- [tex]\(a = 1\)[/tex]
- [tex]\(b = 3\)[/tex]
- [tex]\(c = -6\)[/tex]
The Quadratic Formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Let's break down the steps to solve for the roots:
1. Calculate the Discriminant:
The discriminant ([tex]\( \Delta \)[/tex]) is found using the formula:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 3^2 - 4 \cdot 1 \cdot (-6) = 9 + 24 = 33 \][/tex]
2. Check the Sign of the Discriminant:
Since the discriminant is positive ([tex]\( \Delta = 33 \)[/tex]), this indicates that there are two real and distinct roots.
3. Calculate the Roots:
Using the Quadratic Formula, we find the roots as follows:
[tex]\[ x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x_{1,2} = \frac{-3 \pm \sqrt{33}}{2 \cdot 1} \][/tex]
Thus:
- The first root ([tex]\(x_1\)[/tex]) is:
[tex]\[ x_1 = \frac{-3 + \sqrt{33}}{2} \approx 1.3722813232690143 \][/tex]
- The second root ([tex]\(x_2\)[/tex]) is:
[tex]\[ x_2 = \frac{-3 - \sqrt{33}}{2} \approx -4.372281323269014 \][/tex]
Summarizing, the quadratic equation [tex]\(x^2 + 3x - 6 = 0\)[/tex] has the discriminant 33, and the roots are approximately 1.3722813232690143 and -4.372281323269014.
