Simplify The Expression:$z^2 - 12z + 20$

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Introduction

In algebra, simplifying expressions is a crucial step in solving equations and inequalities. It involves rewriting an expression in a more compact and manageable form, often by combining like terms or factoring out common factors. In this article, we will focus on simplifying the given quadratic expression $z^2 - 12z + 20$. We will explore various methods to simplify this expression, including factoring, completing the square, and using the quadratic formula.

Understanding the Expression

Before we begin simplifying the expression, let's take a closer look at its structure. The given expression is a quadratic expression in the form of $az^2 + bz + c$, where $a = 1$, $b = -12$, and $c = 20$. This type of expression can be factored, completed, or solved using the quadratic formula.

Factoring the Expression

One way to simplify the expression is by factoring it into the product of two binomials. To do this, we need to find two numbers whose product is $20$ and whose sum is $-12$. These numbers are $-10$ and $-2$, since $(-10) \times (-2) = 20$ and $(-10) + (-2) = -12$. Therefore, we can write the expression as:

$$z^2 - 12z + 20 = (z - 10)(z - 2)$$

This is a factored form of the original expression, where each factor is a binomial.

Completing the Square

Another method to simplify the expression is by completing the square. This involves rewriting the expression in a perfect square trinomial form. To do this, we need to add and subtract a constant term to make the expression a perfect square. In this case, we can add and subtract $36$ to the expression:

$$z^2 - 12z + 20 = (z^2 - 12z + 36) - 16$$

Now, we can rewrite the expression as:

$$(z - 6)^2 - 16$$

This is a simplified form of the original expression, where the expression is written as a perfect square trinomial.

Using the Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation in the form of $az^2 + bz + c = 0$, the solutions are given by:

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

In this case, we have $a = 1$, $b = -12$, and $c = 20$. Plugging these values into the quadratic formula, we get:

$$z = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(1)(20)}}{2(1)}$$

Simplifying the expression, we get:

$$z = \frac{12 \pm \sqrt{144 - 80}}{2}$$

$$z = \frac{12 \pm \sqrt{64}}{2}$$

$$z = \frac{12 \pm 8}{2}$$

Therefore, the solutions to the equation are:

$$z = \frac{12 + 8}{2} = 10$$

$$z = \frac{12 - 8}{2} = 2$$

Conclusion

In this article, we simplified the quadratic expression $z^2 - 12z + 20$ using various methods, including factoring, completing the square, and using the quadratic formula. We found that the expression can be factored into the product of two binomials, rewritten as a perfect square trinomial, or solved using the quadratic formula. Each method provides a different perspective on simplifying the expression, and understanding these methods is essential for solving quadratic equations and inequalities.

Final Answer

The final answer to the problem is:

$$z^2 - 12z + 20 = (z - 10)(z - 2) = (z - 6)^2 - 16 = \frac{12 \pm 8}{2}$$

The solutions to the equation are $z = 10$ and $z = 2$.

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Introduction

In our previous article, we simplified the quadratic expression $z^2 - 12z + 20$ using various methods, including factoring, completing the square, and using the quadratic formula. In this article, we will address some common questions and concerns related to simplifying quadratic expressions.

Q: What is the difference between factoring and completing the square?

A: Factoring and completing the square are two different methods for simplifying quadratic expressions. Factoring involves rewriting the expression as the product of two binomials, while completing the square involves rewriting the expression as a perfect square trinomial.

Q: How do I know which method to use?

A: The choice of method depends on the specific expression and the desired outcome. If the expression can be easily factored, then factoring may be the best approach. If the expression is not easily factored, then completing the square or using the quadratic formula may be more suitable.

Q: Can I use the quadratic formula to solve any quadratic equation?

A: Yes, the quadratic formula can be used to solve any quadratic equation in the form of $az^2 + bz + c = 0$. However, the formula may not always yield a real solution, especially if the discriminant ($b^2 - 4ac$) is negative.

Q: What is the discriminant, and why is it important?

A: The discriminant is the expression $b^2 - 4ac$ in the quadratic formula. It determines the nature of the solutions to the equation. If the discriminant is positive, then the equation has two real solutions. If the discriminant is zero, then the equation has one real solution. If the discriminant is negative, then the equation has no real solutions.

Q: Can I use the quadratic formula to solve quadratic expressions with complex coefficients?

A: Yes, the quadratic formula can be used to solve quadratic expressions with complex coefficients. However, the solutions will be complex numbers, rather than real numbers.

Q: How do I simplify a quadratic expression with complex coefficients?

A: To simplify a quadratic expression with complex coefficients, you can use the same methods as for real coefficients, including factoring, completing the square, and using the quadratic formula. However, you will need to be careful when working with complex numbers and their conjugates.

Q: Can I use the quadratic formula to solve quadratic expressions with rational coefficients?

A: Yes, the quadratic formula can be used to solve quadratic expressions with rational coefficients. However, the solutions may not always be rational numbers, especially if the discriminant is not a perfect square.

Q: How do I simplify a quadratic expression with rational coefficients?

A: To simplify a quadratic expression with rational coefficients, you can use the same methods as for real coefficients, including factoring, completing the square, and using the quadratic formula. However, you will need to be careful when working with rational numbers and their fractions.

Conclusion

In this article, we addressed some common questions and concerns related to simplifying quadratic expressions. We discussed the differences between factoring and completing the square, the choice of method, and the use of the quadratic formula. We also explored the use of the quadratic formula for solving quadratic expressions with complex and rational coefficients.

Final Answer

The final answer to the problem is:

$$z^2 - 12z + 20 = (z - 10)(z - 2) = (z - 6)^2 - 16 = \frac{12 \pm 8}{2}$$

The solutions to the equation are $z = 10$ and $z = 2$.