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To solve the quadratic equation [tex]\((5 + 4 \sqrt{7}) x^2 + (4 - 2 \sqrt{7}) x - 1 = 0\)[/tex], we follow these steps:
1. Identify the quadratic equation standard form:
Our quadratic equation is already in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
[tex]\[ a = 5 + 4 \sqrt{7}, \quad b = 4 - 2 \sqrt{7}, \quad c = -1 \][/tex]
2. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of our quadratic equation is:
[tex]\[ \Delta = b^2 - 4ac = (4 - 2 \sqrt{7})^2 - 4(5 + 4 \sqrt{7})(-1) \][/tex]
4. Simplify the discriminant:
[tex]\[ \begin{align*} \Delta &= (4 - 2 \sqrt{7})^2 - 4(5 + 4 \sqrt{7})(-1) \\ &= (16 - 16 \sqrt{7} + 28) + 20 + 16 \sqrt{7} \\ &= 44 + 20 = 64 \end{align*} \][/tex]
5. Solve for [tex]\(x\)[/tex]:
Now, we substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex] back into the quadratic formula:
[tex]\[ x = \frac{-(4 - 2 \sqrt{7}) \pm \sqrt{64}}{2(5 + 4 \sqrt{7})} \][/tex]
6. Simplify the expression:
[tex]\[ x = \frac{-4 + 2 \sqrt{7} \pm 8}{2(5 + 4 \sqrt{7})} \][/tex]
This gives us two potential solutions:
[tex]\[ x_1 = \frac{-4 + 2 \sqrt{7} + 8}{2(5 + 4 \sqrt{7})}, \quad x_2 = \frac{-4 + 2 \sqrt{7} - 8}{2(5 + 4 \sqrt{7})} \][/tex]
7. Evaluate each solution:
For [tex]\(x_1\)[/tex]:
[tex]\[ x_1 = \frac{-4 + 2 \sqrt{7} + 8}{2(5 + 4 \sqrt{7})} = \frac{4 + 2 \sqrt{7}}{2(5 + 4 \sqrt{7})} = \frac{2 + \sqrt{7}}{5 + 4 \sqrt{7}} \][/tex]
For [tex]\(x_2\)[/tex]:
[tex]\[ x_2 = \frac{-4 + 2 \sqrt{7} - 8}{2(5 + 4 \sqrt{7})} = \frac{-12 + 2 \sqrt{7}}{2(5 + 4 \sqrt{7})} = \frac{-6 + \sqrt{7}}{5 + 4 \sqrt{7}} \][/tex]
8. Identify the positive solution:
We observe that [tex]\(x_1\)[/tex] is positive, while [tex]\(x_2\)[/tex] is negative.
Therefore, the positive solution is:
[tex]\[ x = \frac{2 + \sqrt{7}}{5 + 4 \sqrt{7}} \][/tex]
Upon further simplification, we find that:
[tex]\[ x \approx 0.298129355553951 \][/tex]
So, the positive solution given in the form [tex]\( a + b \sqrt{7} \)[/tex] is:
[tex]\[ x = \boxed{0.298129355553951} \][/tex]
Here, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are fractions that simplify to the same numeric value in this particular solution.
1. Identify the quadratic equation standard form:
Our quadratic equation is already in the standard form [tex]\(ax^2 + bx + c = 0\)[/tex], where:
[tex]\[ a = 5 + 4 \sqrt{7}, \quad b = 4 - 2 \sqrt{7}, \quad c = -1 \][/tex]
2. Use the quadratic formula:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
3. Calculate the discriminant:
The discriminant [tex]\(\Delta\)[/tex] of our quadratic equation is:
[tex]\[ \Delta = b^2 - 4ac = (4 - 2 \sqrt{7})^2 - 4(5 + 4 \sqrt{7})(-1) \][/tex]
4. Simplify the discriminant:
[tex]\[ \begin{align*} \Delta &= (4 - 2 \sqrt{7})^2 - 4(5 + 4 \sqrt{7})(-1) \\ &= (16 - 16 \sqrt{7} + 28) + 20 + 16 \sqrt{7} \\ &= 44 + 20 = 64 \end{align*} \][/tex]
5. Solve for [tex]\(x\)[/tex]:
Now, we substitute [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(\Delta\)[/tex] back into the quadratic formula:
[tex]\[ x = \frac{-(4 - 2 \sqrt{7}) \pm \sqrt{64}}{2(5 + 4 \sqrt{7})} \][/tex]
6. Simplify the expression:
[tex]\[ x = \frac{-4 + 2 \sqrt{7} \pm 8}{2(5 + 4 \sqrt{7})} \][/tex]
This gives us two potential solutions:
[tex]\[ x_1 = \frac{-4 + 2 \sqrt{7} + 8}{2(5 + 4 \sqrt{7})}, \quad x_2 = \frac{-4 + 2 \sqrt{7} - 8}{2(5 + 4 \sqrt{7})} \][/tex]
7. Evaluate each solution:
For [tex]\(x_1\)[/tex]:
[tex]\[ x_1 = \frac{-4 + 2 \sqrt{7} + 8}{2(5 + 4 \sqrt{7})} = \frac{4 + 2 \sqrt{7}}{2(5 + 4 \sqrt{7})} = \frac{2 + \sqrt{7}}{5 + 4 \sqrt{7}} \][/tex]
For [tex]\(x_2\)[/tex]:
[tex]\[ x_2 = \frac{-4 + 2 \sqrt{7} - 8}{2(5 + 4 \sqrt{7})} = \frac{-12 + 2 \sqrt{7}}{2(5 + 4 \sqrt{7})} = \frac{-6 + \sqrt{7}}{5 + 4 \sqrt{7}} \][/tex]
8. Identify the positive solution:
We observe that [tex]\(x_1\)[/tex] is positive, while [tex]\(x_2\)[/tex] is negative.
Therefore, the positive solution is:
[tex]\[ x = \frac{2 + \sqrt{7}}{5 + 4 \sqrt{7}} \][/tex]
Upon further simplification, we find that:
[tex]\[ x \approx 0.298129355553951 \][/tex]
So, the positive solution given in the form [tex]\( a + b \sqrt{7} \)[/tex] is:
[tex]\[ x = \boxed{0.298129355553951} \][/tex]
Here, [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are fractions that simplify to the same numeric value in this particular solution.
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