The problem asks to determine the parameters $p$ and $q$ for each quadratic equation, where the equation is written in the normal form $x^2 + px + q = 0$. We need to rewrite each given equation in this form and then identify the values of $p$ and $q$.

1. Problem Description

The problem asks to determine the parameters $p$ and $q$ for each quadratic equation, where the equation is written in the normal form $x^2 + px + q = 0$. We need to rewrite each given equation in this form and then identify the values of $p$ and $q$.

2. Solution Steps

a) $5x^2 + 15x - 10 = 0$

Divide by 5: $x^2 + 3x - 2 = 0$.

Therefore, $p = 3$ and $q = -2$.

b) $-x^2 + 2.5x - 1.5 = 0$

Multiply by -1: $x^2 - 2.5x + 1.5 = 0$.

Therefore, $p = -2.5$ and $q = 1.5$.

c) $3x - x^2 + 4 = 0$

Rearrange: $-x^2 + 3x + 4 = 0$.

Multiply by -1: $x^2 - 3x - 4 = 0$.

Therefore, $p = -3$ and $q = -4$.

d) $8 - 7x = x^2$

Rearrange: $x^2 + 7x - 8 = 0$.

Therefore, $p = 7$ and $q = -8$.

e) $x^2 + 3 = 4x$

Rearrange: $x^2 - 4x + 3 = 0$.

Therefore, $p = -4$ and $q = 3$.

f) $2x^2 + 6x = 10$

Rearrange: $2x^2 + 6x - 10 = 0$.

Divide by 2: $x^2 + 3x - 5 = 0$.

Therefore, $p = 3$ and $q = -5$.

g) $x(x-5) + 1 = 0$

Expand: $x^2 - 5x + 1 = 0$.

Therefore, $p = -5$ and $q = 1$.

h) $x(2-x) = 0$

Expand: $2x - x^2 = 0$.

Rearrange: $-x^2 + 2x = 0$.

Multiply by -1: $x^2 - 2x = 0$.

Therefore, $p = -2$ and $q = 0$.

i) $4x^2 + 6 = 6x^2 + 4$

Rearrange: $2x^2 - 2 = 0$.

Divide by 2: $x^2 - 1 = 0$.

Therefore, $p = 0$ and $q = -1$.

3. Final Answer

a) $p = 3$, $q = -2$

b) $p = -2.5$, $q = 1.5$

c) $p = -3$, $q = -4$

d) $p = 7$, $q = -8$

e) $p = -4$, $q = 3$

f) $p = 3$, $q = -5$

g) $p = -5$, $q = 1$

h) $p = -2$, $q = 0$

i) $p = 0$, $q = -1$