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Quadratic equations play a huge role in solving number puzzles and many real-world problems. These are equations where the highest exponent of the variable is 2. For example, the standard form is given by: \[ ax^2 + bx + c = 0 \] In our exercise, the puzzles presented by Stephen, Consuela, and Kwame ultimately boil down to solving quadratic equations. Charging forward with that understanding helps us grasp their individual solutions more clearly.

Let's look at Stephen's puzzle first, which results in the equation: \[ (x - 1)^2 + 5 = 4 \] Simplifying, we get:\[ (x - 1)^2 = -1 \] Here, \( (x - 1)^2 \) is a perfect square, and the fact that it equals \( -1 \) tells us there are no real solutions, since a square number cannot be negative.

Conversely, Consuela's puzzle leads us to the equation:\[ (y - 1)^2 + 1 = 1 \] Simplification results in:\[ (y - 1)^2 = 0 \] Thus, we can solve directly to find \( y = 1 \), indicating one solution.

Finally, Kwame's puzzle involves:\[ (2z - 5)^2 + 1 = 10 \] Simplifying, we get:\[ (2z - 5)^2 = 9 \] Leading to two possible solutions due to the quadratic nature: \( 2z - 5 = 3 \) and \( 2z - 5 = -3 \) giving \( z = 4 \) and \( z = 1 \). Understanding these transformations and solving steps is key to mastering quadratic problems!

Solving equations is a fundamental skill in algebra. It involves finding the values for the variables that satisfy the equation given. For each number puzzle in our exercise, solving involves a series of logical steps.

First, we work on isolating the variable term. For example, in Consuela's puzzle:\[ (y - 1)^2 + 1 = 1 \] The first step is to subtract 1 from both sides to get:\[ (y - 1)^2 = 0 \] Then, we take the square root of both sides, leading to the simple solution:\( y = 1 \).

Similarly, Kwame's puzzle requires a few more steps but follows the same principle. Simplify the equation:\[ (2z - 5)^2 + 1 = 10 \] to:\[ (2z - 5)^2 = 9 \] Next, take the square root of both sides: \( 2z - 5 = 3 \) or \( 2z - 5 = -3 \), which then gives the solutions \( z = 4 \) and \( z = 1 \).

Solving equations often involves strategies like factoring, isolating variables, and checking for extraneous solutions. Consistently applying these steps ensures you will reach the right solution every time.

Number puzzles are a fun way to apply mathematical concepts and problem-solving skills. These puzzles often give clues in a veiled form, requiring translation into mathematical equations.

In our example, Stephen's puzzle states: 'If you subtract 1 from my number, square the result, and add 5, you will get 4.' Translating this into an equation:\[ (x - 1)^2 + 5 = 4 \] We end up with \( (x - 1)^2 = -1 \), indicating no possible real number when squared can result in \( -1 \).

For Consuela's puzzle: 'If you subtract 1 from my number, square the result, and add 1, you will get 1.' This translates to:\[ (y - 1)^2 + 1 = 1 \] Simplifying, we see:\[ (y - 1)^2 = 0 \], so \( y = 1 \), showing a unique solution.

Kwame鈥檚 description: 'If you double the number, subtract 5, square the result, and add 1, you will get 10.' Converting this, we get:\[ (2z - 5)^2 + 1 = 10 \] Simplifying, we have:\[ (2z - 5)^2 = 9 \]. This leads to two potential solutions:\( 2z - 5 = 3 \) and \( 2z - 5 = -3 \), making \( z = 4 \) and \( z = 1 \).

Through translating words into math equations and solving each step meticulously, number puzzles foster critical thinking and algebra skills that are incredibly useful.