Answer :
To determine the correct substitution of the values \(a = -3\), \(b = -2\), and \(c = 6\) into the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), we need to substitute these values step-by-step. Let's analyze each option to see which one is correctly substituted.
### Quadratic Formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Substituting Values:
The given equation is:
[tex]\[ 0 = -3x^2 - 2x + 6 \][/tex]
Here, \( a = -3 \), \( b = -2 \), and \( c = 6 \).
To substitute these into the quadratic formula, we follow these steps:
1. Substitute \( b = -2 \):
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4ac}}{2a} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 - 4ac}}{2a} \][/tex]
2. Substitute \( a = -3 \):
[tex]\[ x = \frac{2 \pm \sqrt{4 - 4(-3)c}}{2(-3)} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 12c}}{-6} \][/tex]
3. Substitute \( c = 6 \):
[tex]\[ x = \frac{2 \pm \sqrt{4 + 12 \cdot 6}}{-6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 72}}{-6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{76}}{-6} \][/tex]
### Evaluating the Given Options:
From the provided options, let's find the one that matches this substitution process:
1. \( x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \)
2. \( x = \frac{-2 \pm \sqrt{2^2 - 4(-3)(6)}}{2(-3)} \)
3. \( x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(6)}}{2(3)} \)
4. \( x = \frac{-2 \pm \sqrt{2^2 - 4(3)(6)}}{2(3)} \)
### Analyzing Each Option:
1. Substituting correctly:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
This equals:
[tex]\[ x = \frac{2 \pm \sqrt{4 + 72}}{-6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{76}}{-6} \][/tex]
This option matches our steps.
2. Incorrect because \( -2 \) is used instead of \( -(-2) \):
[tex]\[ x = \frac{-2 \pm \sqrt{2^2 - 4(-3)(6)}}{2(-3)} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{4 + 72}}{-6} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{76}}{-6} \][/tex]
3. Incorrect because \( a = 3 \) instead of \( a = -3 \):
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(6)}}{2(3)} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 - 72}}{6} \][/tex]
[tex]\[ x \neq \frac{2 \pm \sqrt{76}}{-6} \][/tex]
4. Incorrect because both \( b \) and \( a \) are mis-used:
[tex]\[ x = \frac{-2 \pm \sqrt{2^2 - 4(3)(6)}}{2(3)} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{4 - 72}}{6} \][/tex]
### Conclusion:
The correct substitution of the values \( a = -3 \), \( b = -2 \), and \( c = 6 \) into the quadratic formula is:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
So, option 1 is the correct substitution.
### Quadratic Formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Substituting Values:
The given equation is:
[tex]\[ 0 = -3x^2 - 2x + 6 \][/tex]
Here, \( a = -3 \), \( b = -2 \), and \( c = 6 \).
To substitute these into the quadratic formula, we follow these steps:
1. Substitute \( b = -2 \):
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4ac}}{2a} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 - 4ac}}{2a} \][/tex]
2. Substitute \( a = -3 \):
[tex]\[ x = \frac{2 \pm \sqrt{4 - 4(-3)c}}{2(-3)} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 12c}}{-6} \][/tex]
3. Substitute \( c = 6 \):
[tex]\[ x = \frac{2 \pm \sqrt{4 + 12 \cdot 6}}{-6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 + 72}}{-6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{76}}{-6} \][/tex]
### Evaluating the Given Options:
From the provided options, let's find the one that matches this substitution process:
1. \( x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \)
2. \( x = \frac{-2 \pm \sqrt{2^2 - 4(-3)(6)}}{2(-3)} \)
3. \( x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(6)}}{2(3)} \)
4. \( x = \frac{-2 \pm \sqrt{2^2 - 4(3)(6)}}{2(3)} \)
### Analyzing Each Option:
1. Substituting correctly:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
This equals:
[tex]\[ x = \frac{2 \pm \sqrt{4 + 72}}{-6} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{76}}{-6} \][/tex]
This option matches our steps.
2. Incorrect because \( -2 \) is used instead of \( -(-2) \):
[tex]\[ x = \frac{-2 \pm \sqrt{2^2 - 4(-3)(6)}}{2(-3)} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{4 + 72}}{-6} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{76}}{-6} \][/tex]
3. Incorrect because \( a = 3 \) instead of \( a = -3 \):
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(3)(6)}}{2(3)} \][/tex]
[tex]\[ x = \frac{2 \pm \sqrt{4 - 72}}{6} \][/tex]
[tex]\[ x \neq \frac{2 \pm \sqrt{76}}{-6} \][/tex]
4. Incorrect because both \( b \) and \( a \) are mis-used:
[tex]\[ x = \frac{-2 \pm \sqrt{2^2 - 4(3)(6)}}{2(3)} \][/tex]
[tex]\[ x = \frac{-2 \pm \sqrt{4 - 72}}{6} \][/tex]
### Conclusion:
The correct substitution of the values \( a = -3 \), \( b = -2 \), and \( c = 6 \) into the quadratic formula is:
[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(-3)(6)}}{2(-3)} \][/tex]
So, option 1 is the correct substitution.
