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Quadratic equations are a type of polynomial equation of the second degree. They have the general form: \[ ax^2 + bx + c = 0 \] Here, *a*, *b*, and *c* are constants with *a* ≠ 0. Quadratic equations are characterized by the highest exponent of the variable being 2. Some common methods to solve quadratic equations include:

• Factoring
• Using the quadratic formula
• Completing the square

Completing the square involves converting the quadratic equation into a perfect square trinomial. This method is particularly useful when the quadratic equation does not factor nicely. It also sets the stage for deriving the quadratic formula.

Let's discuss how to solve quadratic equations by completing the square, using the given example \[ t^2 - 10t = -23 \]. Here's how it works:First, we move the constant term to the other side of the equation. \[ t^2 - 10t + 23 = 0 \] Next, focus on the linear term. The coefficient of *t* is -10. Take half of this coefficient, square it, and add and subtract this value within the equation.

• Half of -10 is -5.
• Square -5 to get 25.

So, we rewrite our equation adding and subtracting 25:\[ t^2 - 10t + 25 - 25 = 23 \]This simplifies to: \[ t^2 - 10t + 25 = 48 \]The left-hand side of the equation is now a perfect square trinomial. Therefore, we can write it as: \[ (t - 5)^2 = 48 \] Finally, solve for *t* by taking the square root of both sides and isolating *t*. Don't forget to consider both the positive and negative roots: \[ t - 5 = \pm 4\sqrt{3} \] Thus, \[ t = 5 \pm 4\sqrt{3} \].

Algebraic techniques are fundamental tools used to manipulate and solve equations. These include:

• Rearranging equations
• Combining like terms
• Using distributive property
• Factoring
• Expanding

When solving quadratic equations by completing the square, we utilize several of these techniques. Here are some key steps and the techniques involved:

• Adding or subtracting the same quantity to both sides of the equation (rearranging equations)
• Identifying and manipulating coefficients
• Using the property \[(a - b)^2 = a^2 - 2ab + b^2\] to form a perfect square trinomial
• Factoring the trinomial into a binomial square
• Solving the resulting simplified balanced equation

Understanding these techniques builds a foundation for tackling more complex algebraic problems. It's essential to practice these to achieve mastery in algebra.