Get reliable answers to your questions at Westonci.ca, where our knowledgeable community is always ready to help. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Sure, let's break down the solution step-by-step and identify any errors.
1. Starting Equation:
[tex]\[ 9x + 2 = 8x^2 + 6x \][/tex]
2. Rearrange to Standard Form:
Move all terms to one side to form a quadratic equation in standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[ -8x^2 + 3x + 2 = 0 \][/tex]
3. Identify Coefficients:
The quadratic equation in standard form is:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Here, [tex]\( a = -8 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 2 \)[/tex].
4. Quadratic Formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
5. Compute the Discriminant:
The discriminant of the quadratic equation is:
[tex]\[ b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ b^2 - 4ac = 3^2 - 4(-8)(2) = 9 + 64 = 73 \][/tex]
6. Apply the Quadratic Formula:
Since the discriminant is positive (73), the roots are real and distinct. Substitute the values into the quadratic formula:
[tex]\[ x = \frac{-3 \pm \sqrt{73}}{-16} \][/tex]
Simplify the expression for the roots:
[tex]\[ x = \frac{-3 + \sqrt{73}}{-16} \quad \text{and} \quad x = \frac{-3 - \sqrt{73}}{-16} \][/tex]
Which simplifies to:
[tex]\[ x_1 = \frac{3 - \sqrt{73}}{16} \quad \text{and} \quad x_2 = \frac{3 + \sqrt{73}}{16} \][/tex]
Evaluating these numerically:
- Calculate [tex]\(x_1 \approx -0.3465\)[/tex]
- Calculate [tex]\(x_2 \approx 0.7215\)[/tex]
7. Numerical Results:
The numerical solution for the roots of the quadratic equation is:
[tex]\[ x_1 \approx -0.3465 \quad \text{and} \quad x_2 \approx 0.7215 \][/tex]
### Comparison with Initial Solution:
- The initial solution contained a mistake in handling the discriminant:
[tex]\[ \frac{-3 \pm \sqrt{9 - 64i}}{-16} \rightarrow \frac{3 \pm \sqrt{55i}}{16} \][/tex]
This interpretation was incorrect since the discriminant, when evaluated properly (as [tex]\(73\)[/tex]), yields real numbers as roots.
### Conclusion:
Given all the steps correctly, the final step confirms that the roots of the equation [tex]\( -8x^2 + 3x + 2 = 0 \)[/tex] are indeed real numbers and approximately:
[tex]\[ x_1 \approx -0.3465 \quad \text{and} \quad x_2 \approx 0.7215 \][/tex]
The initial solution improperly concluded complex roots. Therefore, the correct final numerical results are real and are:
[tex]\[ -0.3465 \quad \text{and} \quad 0.7215 \][/tex]
1. Starting Equation:
[tex]\[ 9x + 2 = 8x^2 + 6x \][/tex]
2. Rearrange to Standard Form:
Move all terms to one side to form a quadratic equation in standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]:
[tex]\[ -8x^2 + 3x + 2 = 0 \][/tex]
3. Identify Coefficients:
The quadratic equation in standard form is:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
Here, [tex]\( a = -8 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 2 \)[/tex].
4. Quadratic Formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
5. Compute the Discriminant:
The discriminant of the quadratic equation is:
[tex]\[ b^2 - 4ac \][/tex]
Substituting the values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ b^2 - 4ac = 3^2 - 4(-8)(2) = 9 + 64 = 73 \][/tex]
6. Apply the Quadratic Formula:
Since the discriminant is positive (73), the roots are real and distinct. Substitute the values into the quadratic formula:
[tex]\[ x = \frac{-3 \pm \sqrt{73}}{-16} \][/tex]
Simplify the expression for the roots:
[tex]\[ x = \frac{-3 + \sqrt{73}}{-16} \quad \text{and} \quad x = \frac{-3 - \sqrt{73}}{-16} \][/tex]
Which simplifies to:
[tex]\[ x_1 = \frac{3 - \sqrt{73}}{16} \quad \text{and} \quad x_2 = \frac{3 + \sqrt{73}}{16} \][/tex]
Evaluating these numerically:
- Calculate [tex]\(x_1 \approx -0.3465\)[/tex]
- Calculate [tex]\(x_2 \approx 0.7215\)[/tex]
7. Numerical Results:
The numerical solution for the roots of the quadratic equation is:
[tex]\[ x_1 \approx -0.3465 \quad \text{and} \quad x_2 \approx 0.7215 \][/tex]
### Comparison with Initial Solution:
- The initial solution contained a mistake in handling the discriminant:
[tex]\[ \frac{-3 \pm \sqrt{9 - 64i}}{-16} \rightarrow \frac{3 \pm \sqrt{55i}}{16} \][/tex]
This interpretation was incorrect since the discriminant, when evaluated properly (as [tex]\(73\)[/tex]), yields real numbers as roots.
### Conclusion:
Given all the steps correctly, the final step confirms that the roots of the equation [tex]\( -8x^2 + 3x + 2 = 0 \)[/tex] are indeed real numbers and approximately:
[tex]\[ x_1 \approx -0.3465 \quad \text{and} \quad x_2 \approx 0.7215 \][/tex]
The initial solution improperly concluded complex roots. Therefore, the correct final numerical results are real and are:
[tex]\[ -0.3465 \quad \text{and} \quad 0.7215 \][/tex]
Thank you for your visit. We are dedicated to helping you find the information you need, whenever you need it. Thank you for your visit. We're dedicated to helping you find the information you need, whenever you need it. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
