Which Function Is Equivalent To $q(x) = 9x^2 - 24x + 16$?

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Introduction to Quadratic Functions

Quadratic functions are a fundamental concept in mathematics, and they play a crucial role in various fields, including algebra, geometry, and calculus. A quadratic function is a polynomial function of degree two, which means the highest power of the variable is two. The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be zero.

In this article, we will explore the function $q(x) = 9x^2 - 24x + 16$ and determine which function is equivalent to it. We will use various techniques, including factoring, completing the square, and the quadratic formula, to find the equivalent function.

Factoring the Quadratic Function

One way to find the equivalent function is to factor the quadratic expression. Factoring involves expressing the quadratic expression as a product of two binomials. To factor the quadratic expression $9x^2 - 24x + 16$, we need to find two numbers whose product is $9 \times 16 = 144$ and whose sum is $-24$. These numbers are $-18$ and $-8$, so we can write the quadratic expression as:

$$9x^2 - 24x + 16 = (3x - 4)(3x - 4)$$

However, this is not the only way to factor the quadratic expression. We can also write it as:

$$9x^2 - 24x + 16 = (3x - 2)(3x - 8)$$

Completing the Square

Another way to find the equivalent function is to complete the square. Completing the square involves expressing the quadratic expression in the form $(x + d)^2 + e$, where $d$ and $e$ are constants. To complete the square, we need to find the value of $d$ such that the quadratic expression can be written in the form $(x + d)^2 + e$. In this case, we can write the quadratic expression as:

$$9x^2 - 24x + 16 = (3x - 4)^2 + 0$$

However, this is not the only way to complete the square. We can also write the quadratic expression as:

$$9x^2 - 24x + 16 = (3x - 2)^2 - 4$$

The Quadratic Formula

The quadratic formula is a powerful tool for finding the roots of a quadratic equation. The quadratic formula states that the roots of the quadratic equation $ax^2 + bx + c = 0$ are given by:

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

In this case, we have the quadratic equation $9x^2 - 24x + 16 = 0$. We can use the quadratic formula to find the roots of this equation:

$$x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4 \times 9 \times 16}}{2 \times 9}$$

Simplifying this expression, we get:

$$x = \frac{24 \pm \sqrt{576 - 576}}{18}$$

$$x = \frac{24 \pm \sqrt{0}}{18}$$

$$x = \frac{24}{18}$$

$$x = \frac{4}{3}$$

Conclusion

In this article, we have explored the function $q(x) = 9x^2 - 24x + 16$ and determined which function is equivalent to it. We have used various techniques, including factoring, completing the square, and the quadratic formula, to find the equivalent function. We have shown that the function $q(x) = 9x^2 - 24x + 16$ is equivalent to the function $f(x) = (3x - 4)^2$ and $f(x) = (3x - 2)^2 - 4$. We have also shown that the roots of the quadratic equation $9x^2 - 24x + 16 = 0$ are given by $x = \frac{4}{3}$.

Final Answer

The final answer is $\boxed{(3x - 4)^2}$ and $\boxed{(3x - 2)^2 - 4}$

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Introduction

In our previous article, we explored the function $q(x) = 9x^2 - 24x + 16$ and determined which function is equivalent to it. We used various techniques, including factoring, completing the square, and the quadratic formula, to find the equivalent function. In this article, we will answer some of the most frequently asked questions about the function $q(x) = 9x^2 - 24x + 16$.

Q: What is the general form of a quadratic function?

A: The general form of a quadratic function is $f(x) = ax^2 + bx + c$, where $a$, $b$, and $c$ are constants, and $a$ cannot be zero.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is the constant term and whose sum is the coefficient of the linear term. For example, to factor the quadratic expression $9x^2 - 24x + 16$, you need to find two numbers whose product is $9 \times 16 = 144$ and whose sum is $-24$. These numbers are $-18$ and $-8$, so you can write the quadratic expression as $(3x - 4)(3x - 4)$ or $(3x - 2)(3x - 8)$.

Q: How do I complete the square?

A: To complete the square, you need to find the value of $d$ such that the quadratic expression can be written in the form $(x + d)^2 + e$. For example, to complete the square for the quadratic expression $9x^2 - 24x + 16$, you can write it as $(3x - 4)^2 + 0$ or $(3x - 2)^2 - 4$.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for finding the roots of a quadratic equation. The quadratic formula states that the roots of the quadratic equation $ax^2 + bx + c = 0$ are given by:

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

Q: How do I use the quadratic formula to find the roots of a quadratic equation?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula. For example, to find the roots of the quadratic equation $9x^2 - 24x + 16 = 0$, you can plug in the values $a = 9$, $b = -24$, and $c = 16$ into the formula:

$$x = \frac{-(-24) \pm \sqrt{(-24)^2 - 4 \times 9 \times 16}}{2 \times 9}$$

Simplifying this expression, you get:

$$x = \frac{24 \pm \sqrt{576 - 576}}{18}$$

$$x = \frac{24 \pm \sqrt{0}}{18}$$

$$x = \frac{24}{18}$$

$$x = \frac{4}{3}$$

Q: What are the roots of the quadratic equation $9x^2 - 24x + 16 = 0$?

A: The roots of the quadratic equation $9x^2 - 24x + 16 = 0$ are given by $x = \frac{4}{3}$.

Conclusion

In this article, we have answered some of the most frequently asked questions about the function $q(x) = 9x^2 - 24x + 16$. We have provided explanations and examples to help you understand the concepts and techniques used to find the equivalent function. We hope that this article has been helpful in clarifying any doubts you may have had about the function $q(x) = 9x^2 - 24x + 16$.

Final Answer

The final answer is $\boxed{(3x - 4)^2}$ and $\boxed{(3x - 2)^2 - 4}$