To write the quadratic equation given the roots of the equation, we use the fact that if $r_1$ and $r_2$ are the roots, the quadratic equation can be written as:

$(x - r_1)(x - r_2) = 0$

Expanding this, the quadratic equation becomes:

$x^2 - (r_1 + r_2)x + r_1r_2 = 0$

Now, let's apply this to each of the given pairs of roots.

1. Roots: 8 and 2

• Sum of roots $r_1 + r_2 = 8 + 2 = 10$
• Product of roots $r_1r_2 = 8 \times 2 = 16$

The quadratic equation is:

$x^2 - 10x + 16 = 0$

2. Roots: -3 and 7

• Sum of roots $r_1 + r_2 = -3 + 7 = 4$
• Product of roots $r_1r_2 = -3 \times 7 = -21$

The quadratic equation is:

$x^2 - 4x - 21 = 0$

3. Roots: $\frac{2}{3}$ and $-6$

• Sum of roots $r_1 + r_2 = \frac{2}{3} + (-6) = \frac{2}{3} - \frac{18}{3} = \frac{-16}{3}$
• Product of roots $r_1r_2 = \frac{2}{3} \times (-6) = -4$

The quadratic equation is:

$x^2 + \frac{16}{3}x - 4 = 0$

If you prefer the equation without fractions, multiply the entire equation by 3:

$3x^2 + 16x - 12 = 0$

4. Roots: -3 and -5

• Sum of roots $r_1 + r_2 = -3 + (-5) = -8$
• Product of roots $r_1r_2 = -3 \times -5 = 15$

The quadratic equation is:

$x^2 + 8x + 15 = 0$

Would you like any further details or clarifications?

Here are 5 related questions:

1. How can you derive the quadratic formula from a general quadratic equation?
2. What is the discriminant, and how does it relate to the nature of the roots?
3. How can you determine the axis of symmetry for a quadratic equation?
4. What is the significance of the vertex in a quadratic equation?
5. How can quadratic equations be solved using the method of completing the square?

Tip: Always check the sum and product of the roots when writing the quadratic equation to ensure accuracy.