Solve \[$(2k + 3)^2 - 24 = 3\$\].

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $(2k + 3)^2 - 24 = 3$, using a step-by-step approach. We will break down the solution into manageable parts, making it easier to understand and follow.

Understanding the Equation

The given equation is $(2k + 3)^2 - 24 = 3$. To solve this equation, we need to isolate the variable $k$. The first step is to simplify the equation by expanding the squared term.

Expanding the Squared Term

To expand the squared term, we need to apply the formula $(a + b)^2 = a^2 + 2ab + b^2$. In this case, $a = 2k$ and $b = 3$.

(2k + 3)^2 = (2k)^2 + 2(2k)(3) + 3^2
(2k + 3)^2 = 4k^2 + 12k + 9

Now, we can substitute this expression back into the original equation.

Substituting the Expanded Term

(4k^2 + 12k + 9) - 24 = 3

Next, we need to simplify the equation by combining like terms.

Simplifying the Equation

4k^2 + 12k - 15 = 3

Now, we can move all terms to one side of the equation to set it equal to zero.

Setting the Equation Equal to Zero

4k^2 + 12k - 18 = 0

This is a quadratic equation in the form $ax^2 + bx + c = 0$, where $a = 4$, $b = 12$, and $c = -18$. We can solve this equation using the quadratic formula.

Solving the Quadratic Equation

The quadratic formula is given by:

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

In this case, $a = 4$, $b = 12$, and $c = -18$. Plugging these values into the formula, we get:

k = \frac{-12 \pm \sqrt{12^2 - 4(4)(-18)}}{2(4)}
k = \frac{-12 \pm \sqrt{144 + 288}}{8}
k = \frac{-12 \pm \sqrt{432}}{8}
k = \frac{-12 \pm 12\sqrt{3}}{8}
k = \frac{-3 \pm 3\sqrt{3}}{2}

Therefore, the solutions to the equation $(2k + 3)^2 - 24 = 3$ are $k = \frac{-3 + 3\sqrt{3}}{2}$ and $k = \frac{-3 - 3\sqrt{3}}{2}$.

Conclusion

Solving quadratic equations can be a challenging task, but with the right approach and techniques, it can be made easier. In this article, we solved the equation $(2k + 3)^2 - 24 = 3$ using a step-by-step approach, starting from expanding the squared term to solving the quadratic equation using the quadratic formula. We hope that this article has provided a clear and concise guide to solving quadratic equations, and we encourage readers to practice solving more equations to become proficient in this skill.

Additional Resources

For more information on solving quadratic equations, we recommend the following resources:

• Khan Academy: Quadratic Equations
• Mathway: Quadratic Equation Solver
• Wolfram Alpha: Quadratic Equation Solver

Frequently Asked Questions

Q: What is a quadratic equation? A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two.

Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you can use the quadratic formula, which is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: What is the quadratic formula? A: The quadratic formula is a mathematical formula that gives the solutions to a quadratic equation in the form $ax^2 + bx + c = 0$.

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Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will provide answers to frequently asked questions about quadratic equations, including their definition, how to solve them, and more.

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. It is typically written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula, which is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. You can also use factoring, completing the square, or graphing to solve quadratic equations.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that gives the solutions to a quadratic equation in the form $ax^2 + bx + c = 0$. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ into the formula and simplify the expression. Make sure to follow the order of operations (PEMDAS) and simplify the square root expression.

Q: What is the difference between a quadratic equation and a linear equation?

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. A quadratic equation has a squared term, while a linear equation does not.

Q: Can I solve a quadratic equation by factoring?

A: Yes, you can solve a quadratic equation by factoring if it can be written in the form $(x - r)(x - s) = 0$, where $r$ and $s$ are the roots of the equation.

Q: What is the relationship between the roots of a quadratic equation and its coefficients?

A: The roots of a quadratic equation are related to its coefficients by the quadratic formula. The sum of the roots is equal to $-\frac{b}{a}$, and the product of the roots is equal to $\frac{c}{a}$.

Q: Can I solve a quadratic equation by completing the square?

A: Yes, you can solve a quadratic equation by completing the square if it can be written in the form $ax^2 + bx + c = 0$. This method involves rewriting the equation in the form $(x + \frac{b}{2a})^2 = \frac{c}{a} - \frac{b^2}{4a^2}$.

Q: What is the significance of the discriminant in a quadratic equation?

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

Q: Can I solve a quadratic equation using graphing?

A: Yes, you can solve a quadratic equation using graphing. This involves graphing the equation on a coordinate plane and finding the points where the graph intersects the x-axis. These points represent the roots of the equation.

Conclusion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we provided answers to frequently asked questions about quadratic equations, including their definition, how to solve them, and more. We hope that this article has provided a clear and concise guide to quadratic equations and has helped to clarify any confusion.