We need to solve three problems: 34. Solve the linear-quadratic system of equations: $y = x^2 + 3x - 1$ $y = x + 2$ 35. Solve the linear-quadratic system of equations: $y = -x^2 - 4x + 2$ $y = -x - 2$ 36. Solve the quadratic system of equations: $y = x^2 + 18x + 35$ $y = -x^2 + 2x + 5$

1. Problem Description

We need to solve three problems:

3

4. Solve the linear-quadratic system of equations:

$y = x^2 + 3x - 1$

$y = x + 2$

3

5. Solve the linear-quadratic system of equations:

$y = -x^2 - 4x + 2$

$y = -x - 2$

3

6. Solve the quadratic system of equations:

$y = x^2 + 18x + 35$

$y = -x^2 + 2x + 5$

2. Solution Steps

3

4. Set the two equations equal to each other:

$x^2 + 3x - 1 = x + 2$

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

Factor the quadratic equation:

$(x+3)(x-1) = 0$

So, $x = -3$ or $x = 1$.

If $x = -3$, then $y = -3 + 2 = -1$.

If $x = 1$, then $y = 1 + 2 = 3$.

3

5. Set the two equations equal to each other:

$-x^2 - 4x + 2 = -x - 2$

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

Factor the quadratic equation:

$(x+4)(x-1) = 0$

So, $x = -4$ or $x = 1$.

If $x = -4$, then $y = -(-4) - 2 = 4 - 2 = 2$.

If $x = 1$, then $y = -1 - 2 = -3$.

3

6. Set the two equations equal to each other:

$x^2 + 18x + 35 = -x^2 + 2x + 5$

$2x^2 + 16x + 30 = 0$

Divide by 2:

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

Factor the quadratic equation:

$(x+3)(x+5) = 0$

So, $x = -3$ or $x = -5$.

If $x = -3$, then $y = (-3)^2 + 18(-3) + 35 = 9 - 54 + 35 = -10$.

If $x = -5$, then $y = (-5)^2 + 18(-5) + 35 = 25 - 90 + 35 = -30$.

3. Final Answer

3

4.

$(x, y) = (-3, -1), (1, 3)$

3

5.

$(x, y) = (-4, 2), (1, -3)$

3

6.

$(x, y) = (-3, -10), (-5, -30)$