Discover the answers to your questions at Westonci.ca, where experts share their knowledge and insights with you. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform. Connect with a community of professionals ready to help you find accurate solutions to your questions quickly and efficiently.
Sagot :
To solve the quadratic equation [tex]\(3x^2 + 24x - 24 = 0\)[/tex], let's follow the steps to find the roots step-by-step.
### Step 1: Check that the equation is in standard form
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. In this problem, we have:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 24\)[/tex]
- [tex]\(c = -24\)[/tex]
### Step 2: Use the quadratic formula
The quadratic formula for [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 3: Identify the coefficients
Given:
[tex]\[a = 3\][/tex]
[tex]\[b = 24\][/tex]
[tex]\[c = -24\][/tex]
### Step 4: Calculate the discriminant [tex]\(\Delta\)[/tex]
The discriminant [tex]\(\Delta\)[/tex] is found using the formula [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 24^2 - 4 \cdot 3 \cdot (-24) \][/tex]
[tex]\[ \Delta = 576 + 288 \][/tex]
[tex]\[ \Delta = 864 \][/tex]
### Step 5: Plug the discriminant back into the quadratic formula
Now, substituting [tex]\(\Delta\)[/tex] and the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{-24 \pm \sqrt{864}}{2 \cdot 3} \][/tex]
### Step 6: Simplify under the square root
Recognize that:
[tex]\[ \sqrt{864} = \sqrt{144 \cdot 6} = 12\sqrt{6} \][/tex]
### Step 7: Substitute and simplify
Substitute [tex]\(\sqrt{864} = 12\sqrt{6}\)[/tex] back into the quadratic formula:
[tex]\[ x = \frac{-24 \pm 12\sqrt{6}}{6} \][/tex]
Now, split the two terms:
[tex]\[ x = \frac{-24}{6} \pm \frac{12\sqrt{6}}{6} \][/tex]
[tex]\[ x = -4 \pm 2\sqrt{6} \][/tex]
### Step 8: State the final result
Hence, the solutions to the quadratic equation [tex]\(3x^2 + 24x - 24 = 0\)[/tex] are:
[tex]\[ x = -4 \pm 2\sqrt{6} \][/tex]
Thus, the correct answer choice is:
A. [tex]\(x = -4 \pm 2\sqrt{6}\)[/tex]
### Step 1: Check that the equation is in standard form
The standard form of a quadratic equation is [tex]\(ax^2 + bx + c = 0\)[/tex]. In this problem, we have:
- [tex]\(a = 3\)[/tex]
- [tex]\(b = 24\)[/tex]
- [tex]\(c = -24\)[/tex]
### Step 2: Use the quadratic formula
The quadratic formula for [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 3: Identify the coefficients
Given:
[tex]\[a = 3\][/tex]
[tex]\[b = 24\][/tex]
[tex]\[c = -24\][/tex]
### Step 4: Calculate the discriminant [tex]\(\Delta\)[/tex]
The discriminant [tex]\(\Delta\)[/tex] is found using the formula [tex]\(b^2 - 4ac\)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 24^2 - 4 \cdot 3 \cdot (-24) \][/tex]
[tex]\[ \Delta = 576 + 288 \][/tex]
[tex]\[ \Delta = 864 \][/tex]
### Step 5: Plug the discriminant back into the quadratic formula
Now, substituting [tex]\(\Delta\)[/tex] and the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
[tex]\[ x = \frac{-24 \pm \sqrt{864}}{2 \cdot 3} \][/tex]
### Step 6: Simplify under the square root
Recognize that:
[tex]\[ \sqrt{864} = \sqrt{144 \cdot 6} = 12\sqrt{6} \][/tex]
### Step 7: Substitute and simplify
Substitute [tex]\(\sqrt{864} = 12\sqrt{6}\)[/tex] back into the quadratic formula:
[tex]\[ x = \frac{-24 \pm 12\sqrt{6}}{6} \][/tex]
Now, split the two terms:
[tex]\[ x = \frac{-24}{6} \pm \frac{12\sqrt{6}}{6} \][/tex]
[tex]\[ x = -4 \pm 2\sqrt{6} \][/tex]
### Step 8: State the final result
Hence, the solutions to the quadratic equation [tex]\(3x^2 + 24x - 24 = 0\)[/tex] are:
[tex]\[ x = -4 \pm 2\sqrt{6} \][/tex]
Thus, the correct answer choice is:
A. [tex]\(x = -4 \pm 2\sqrt{6}\)[/tex]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. We're here to help at Westonci.ca. Keep visiting for the best answers to your questions.