We are given the roots of a quadratic equation as:

$r_1 = \frac{3 + \sqrt{7}}{4} \quad \text{and} \quad r_2 = \frac{3 - \sqrt{7}}{4}$

We need to determine which quadratic equation from the options has these roots. We know from Vieta's formulas that for any quadratic equation $ax^2 + bx + c = 0$, the sum and product of the roots are related to the coefficients as follows:

1. Sum of the roots: $r_1 + r_2 = -\frac{b}{a}$
2. Product of the roots: $r_1 \cdot r_2 = \frac{c}{a}$

Step 1: Calculate the sum of the roots.

$r_1 + r_2 = \frac{3 + \sqrt{7}}{4} + \frac{3 - \sqrt{7}}{4} = \frac{(3 + \sqrt{7}) + (3 - \sqrt{7})}{4} = \frac{6}{4} = \frac{3}{2}$

Step 2: Calculate the product of the roots.

$r_1 \cdot r_2 = \left( \frac{3 + \sqrt{7}}{4} \right) \cdot \left( \frac{3 - \sqrt{7}}{4} \right) = \frac{(3 + \sqrt{7})(3 - \sqrt{7})}{16}$ Using the difference of squares formula: $(3 + \sqrt{7})(3 - \sqrt{7}) = 3^2 - (\sqrt{7})^2 = 9 - 7 = 2$ So, $r_1 \cdot r_2 = \frac{2}{16} = \frac{1}{8}$

Step 3: Use the sum and product to find the correct equation.

We now know:

• $r_1 + r_2 = \frac{3}{2}$
• $r_1 \cdot r_2 = \frac{1}{8}$

From Vieta’s relations, we can compare these with the standard form $ax^2 + bx + c = 0$:

• The sum of the roots $-\frac{b}{a} = \frac{3}{2}$
• The product of the roots $\frac{c}{a} = \frac{1}{8}$

Let's check option A: $8x^2 - 12x + 1 = 0$.

• Here, $a = 8$, $b = -12$, and $c = 1$.
• Sum of roots: $-\frac{b}{a} = -\frac{-12}{8} = \frac{12}{8} = \frac{3}{2}$ (matches)
• Product of roots: $\frac{c}{a} = \frac{1}{8}$ (matches)

Thus, option A (8x² - 12x + 1 = 0) is the correct quadratic equation.

Would you like further details or clarifications?

Here are 5 related questions:

1. How do Vieta's formulas work in general for any quadratic equation?
2. How can you derive a quadratic equation from any pair of given roots?
3. How does the discriminant relate to the nature of the roots (real vs. complex)?
4. What is the difference of squares, and how is it applied in simplifying expressions?
5. How do you find the vertex of a quadratic equation using its coefficients?

Tip: Always check both the sum and the product of the roots when determining the correct quadratic equation!