Evaluate z1z2, given that z1 = 2 - 3i and z2 = 5 + 7i.

Step 1: Start with the given equation.
$x^{2} = 4x + 3$

Step 2: Subtract $4x + 3$ from both sides.
$x^{2} - 4x - 3 = 0$

Step 3: Apply the quadratic formula $x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$, where $a=1$, $b=-4$, $c=-3$.
Discriminant: $b^{2} - 4ac = (-4)^{2} - 4(1)(-3) = 16 + 12 = 28$.
$x = \dfrac{4 \pm \sqrt{28}}{2} = \dfrac{4 \pm 2\sqrt{7}}{2} = 2 \pm \sqrt{7}$
x = 2 \pm \sqrt{7}

Step 1: Start with the given equation.
$x^{2} - 5x = -6$

Step 2: Add $6$ to both sides.
$x^{2} - 5x + 6 = 0$

Step 3: Factor the quadratic.
$(x - 2)(x - 3) = 0$

Step 4: Set each factor equal to zero.
$x - 2 = 0 \quad \Rightarrow \quad x = 2$
$x - 3 = 0 \quad \Rightarrow \quad x = 3$
x=2 \quad or \quad x=3

Step 1: Start with the given equation.
$2x^{2} + 5x - 3 = 0$

Step 2: Factor the quadratic.
$(2x - 1)(x + 3) = 0$

Step 3: Set each factor equal to zero.
$2x - 1 = 0 \quad \Rightarrow \quad x = \dfrac{1}{2}$
$x + 3 = 0 \quad \Rightarrow \quad x = -3$
x = \dfrac{1{2} \quad or \quad x = -3}

Step 1: Perform polynomial division of $x^{2} - 4$ by $x - 2$.
Divide leading terms: $\dfrac{x^{2}}{x} = x$.
Multiply: $x(x - 2) = x^{2} - 2x$.
Subtract: $(x^{2} - 4) - (x^{2} - 2x) = 2x - 4$.

Step 2: Divide new leading terms: $\dfrac{2x}{x} = 2$.
Multiply: $2(x - 2) = 2x - 4$.
Subtract: $(2x - 4) - (2x - 4) = 0$.
Quotient is $x + 2$, remainder $0$.
x + 2

Step 1: Start with the given equation.
$x^{2} + 4x + 4 = 0$

Step 2: Recognize as perfect square trinomial.
$(x + 2)^{2} = 0$

Step 3: Solve.
$x + 2 = 0 \quad \Rightarrow \quad x = -2$
x = -2

Step 1: Start with the given equation.
$3x^{2} - 12 = 0$

Step 2: Add $12$ to both sides.
$3x^{2} = 12$

Step 3: Divide by $3$.
$x^{2} = 4$

Step 4: Take square roots.
$x = \pm \sqrt{4} = \pm 2$
x = \pm 2

Step 1: Start with the given equation.
$x^{2} - 4x - 5 = 0$

Step 2: Factor the quadratic.
$(x - 5)(x + 1) = 0$

Step 3: Set each factor equal to zero.
$x - 5 = 0 \quad \Rightarrow \quad x = 5$
$x + 1 = 0 \quad \Rightarrow \quad x = -1$
x = 5 \quad or \quad x = -1

Step 1: Start with the given equation.
$2x^{2} = 8x$

Step 2: Subtract $8x$ from both sides.
$2x^{2} - 8x = 0$

Step 3: Factor out common term.
$2x(x - 4) = 0$

Step 4: Set each factor equal to zero.
$2x = 0 \quad \Rightarrow \quad x = 0$
$x - 4 = 0 \quad \Rightarrow \quad x = 4$
x = 0 \quad or \quad x = 4

Step 1: Start with the given equation.
$x^{2} - 6x + 8 = 0$

Step 2: Factor the quadratic.
$(x - 4)(x - 2) = 0$

Step 3: Set each factor equal to zero.
$x - 4 = 0 \quad \Rightarrow \quad x = 4$
$x - 2 = 0 \quad \Rightarrow \quad x = 2$
x = 4 \quad or \quad x = 2

Step 1: Start with the given equation.
$4x^{2} + 12x + 9 = 0$

Step 2: Recognize as perfect square trinomial.
$(2x + 3)^{2} = 0$

Step 3: Solve.
$2x + 3 = 0 \quad \Rightarrow \quad x = -\dfrac{3}{2}$
x = -\dfrac{3{2}}

Step 1: Start with the given equation.
$x^{2} + 2x - 3 = 0$

Step 2: Factor the quadratic.
$(x + 3)(x - 1) = 0$

Step 3: Set each factor equal to zero.
$x + 3 = 0 \quad \Rightarrow \quad x = -3$
$x - 1 = 0 \quad \Rightarrow \quad x = 1$
x = -3 \quad or \quad x = 1