We are given the equation:

$225 = (61z^2 \times 793) - (4z + 1)^2$

Let's solve for $z$:

Step 1: Simplify the expression

Expand both sides of the equation.

$225 = 61z^2 \times 793 - (4z + 1)^2$

1. First, simplify $61z^2 \times 793$:

$61z^2 \times 793 = 48373z^2$

2. Now expand $(4z + 1)^2$:

$(4z + 1)^2 = (4z)^2 + 2(4z)(1) + 1^2 = 16z^2 + 8z + 1$

Thus, the equation becomes:

$225 = 48373z^2 - (16z^2 + 8z + 1)$

Step 2: Simplify further

Now distribute the negative sign:

$225 = 48373z^2 - 16z^2 - 8z - 1$

Simplify the $z^2$ terms:

$225 = (48373z^2 - 16z^2) - 8z - 1$

$225 = 48357z^2 - 8z - 1$

Step 3: Move all terms to one side

Move all terms to one side of the equation:

$48357z^2 - 8z - 1 - 225 = 0$

Simplify:

$48357z^2 - 8z - 226 = 0$

Step 4: Solve using the quadratic formula

The quadratic formula is:

$z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

For the equation $48357z^2 - 8z - 226 = 0$, we have:

• $a = 48357$
• $b = -8$
• $c = -226$

Now, apply the quadratic formula:

$z = \frac{-(-8) \pm \sqrt{(-8)^2 - 4(48357)(-226)}}{2(48357)}$

Simplify the discriminant:

$z = \frac{8 \pm \sqrt{64 + 4 \times 48357 \times 226}}{2 \times 48357}$

$z = \frac{8 \pm \sqrt{64 + 43733544}}{96714}$

$z = \frac{8 \pm \sqrt{43733608}}{96714}$

$z = \frac{8 \pm 6614.64}{96714}$

Step 5: Solve for both roots

1. $z_1 = \frac{8 + 6614.64}{96714} = \frac{6622.64}{96714} \approx 0.0685$
2. $z_2 = \frac{8 - 6614.64}{96714} = \frac{-6606.64}{96714} \approx -0.0683$

Thus, the two solutions are approximately:

$z \approx 0.0685 \quad \text{or} \quad z \approx -0.0683$

Would you like further details on any step or explanations?

Here are 5 related questions for further exploration:

1. How does the quadratic formula work and why is it useful?
2. What are the steps for solving equations with squared terms?
3. Can this method be used to solve any quadratic equation?
4. How does the discriminant determine the nature of solutions?
5. What are the applications of quadratic equations in real-world problems?

Tip: Always check the discriminant ($b^2 - 4ac$) before applying the quadratic formula. It tells you if the solutions are real or complex.