Get detailed and accurate responses to your questions with IDNLearn.com. Get timely and accurate answers to your questions from our dedicated community of experts who are here to help you.
Sagot :
To solve the quadratic equation [tex]\(3x^2 + 18x + 27 = 0\)[/tex], we follow these steps:
1. Identify the coefficients:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 18\)[/tex]
- [tex]\(c = 27\)[/tex]
2. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 18^2 - 4 \cdot 3 \cdot 27 \][/tex]
[tex]\[ \Delta = 324 - 324 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
3. Interpret the discriminant:
The discriminant being zero ([tex]\(\Delta = 0\)[/tex]) indicates that the quadratic equation has exactly one real solution, also known as a repeated root.
4. Use the quadratic formula:
The solutions of the quadratic equation are given by:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since [tex]\(\Delta = 0\)[/tex], this simplifies to:
[tex]\[ x = \frac{-b}{2a} \][/tex]
5. Calculate the solution:
Substituting the values [tex]\(b = 18\)[/tex] and [tex]\(a = 3\)[/tex]:
[tex]\[ x = \frac{-18}{2 \cdot 3} \][/tex]
[tex]\[ x = \frac{-18}{6} \][/tex]
[tex]\[ x = -3 \][/tex]
Therefore, the solution to the equation [tex]\(3x^2 + 18x + 27 = 0\)[/tex] is:
[tex]\[ x = -3 \][/tex]
Since the discriminant is zero, this means we have a repeated root, so:
[tex]\[ x_1 = x_2 = -3 \][/tex]
1. Identify the coefficients:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 18\)[/tex]
- [tex]\(c = 27\)[/tex]
2. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = 18^2 - 4 \cdot 3 \cdot 27 \][/tex]
[tex]\[ \Delta = 324 - 324 \][/tex]
[tex]\[ \Delta = 0 \][/tex]
3. Interpret the discriminant:
The discriminant being zero ([tex]\(\Delta = 0\)[/tex]) indicates that the quadratic equation has exactly one real solution, also known as a repeated root.
4. Use the quadratic formula:
The solutions of the quadratic equation are given by:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Since [tex]\(\Delta = 0\)[/tex], this simplifies to:
[tex]\[ x = \frac{-b}{2a} \][/tex]
5. Calculate the solution:
Substituting the values [tex]\(b = 18\)[/tex] and [tex]\(a = 3\)[/tex]:
[tex]\[ x = \frac{-18}{2 \cdot 3} \][/tex]
[tex]\[ x = \frac{-18}{6} \][/tex]
[tex]\[ x = -3 \][/tex]
Therefore, the solution to the equation [tex]\(3x^2 + 18x + 27 = 0\)[/tex] is:
[tex]\[ x = -3 \][/tex]
Since the discriminant is zero, this means we have a repeated root, so:
[tex]\[ x_1 = x_2 = -3 \][/tex]
We appreciate every question and answer you provide. Keep engaging and finding the best solutions. This community is the perfect place to learn and grow together. Thank you for choosing IDNLearn.com. We’re here to provide reliable answers, so please visit us again for more solutions.