Solve: $\left(2 A^2 + 5 A + 2\right)^{\frac{1}{2}} = 3$A. $a = -\frac{7}{2}$B. $a = 1$C. $a = -\frac{7}{2}$ Or $a = 1$D. No Real Solution

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 type of quadratic equation, namely the equation involving a square root. We will use the given equation $\left(2 a^2 + 5 a + 2\right)^{\frac{1}{2}} = 3$ as a case study to demonstrate the steps involved in solving such equations.

Understanding the Equation

The given equation involves a square root, which can be written as $\sqrt{2 a^2 + 5 a + 2} = 3$. To solve this equation, we need to isolate the variable $a$ and find its possible values. The first step is to square both sides of the equation to eliminate the square root.

Squaring Both Sides

When we square both sides of the equation, we get:

$$\left(\sqrt{2 a^2 + 5 a + 2}\right)^2 = 3^2$$

This simplifies to:

$$2 a^2 + 5 a + 2 = 9$$

Rearranging the Equation

The next step is to rearrange the equation to form a quadratic equation in the standard form $ax^2 + bx + c = 0$. We can do this by subtracting 9 from both sides of the equation:

$$2 a^2 + 5 a - 7 = 0$$

Solving the Quadratic Equation

Now that we have a quadratic equation in the standard form, we can use various methods to solve it. In this case, we can use the quadratic formula, which is given by:

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

In our case, $a = 2$, $b = 5$, and $c = -7$. Plugging these values into the quadratic formula, we get:

$$a = \frac{-5 \pm \sqrt{5^2 - 4(2)(-7)}}{2(2)}$$

Simplifying this expression, we get:

$$a = \frac{-5 \pm \sqrt{25 + 56}}{4}$$

$$a = \frac{-5 \pm \sqrt{81}}{4}$$

$$a = \frac{-5 \pm 9}{4}$$

Finding the Possible Values of $a$

Now that we have the possible values of $a$, we can simplify the expression to find the final answers. We have two possible values of $a$, namely:

$$a = \frac{-5 + 9}{4} = \frac{4}{4} = 1$$

$$a = \frac{-5 - 9}{4} = \frac{-14}{4} = -\frac{7}{2}$$

Conclusion

In this article, we solved a quadratic equation involving a square root. We used the quadratic formula to find the possible values of the variable $a$. The final answers are $a = 1$ and $a = -\frac{7}{2}$. We hope that this article has provided a clear and step-by-step guide to solving quadratic equations involving square roots.

Answer Key

The correct answer is:

• C. $a = -\frac{7}{2}$ or $a = 1$

Discussion

This problem is a great example of how to solve quadratic equations involving square roots. The key steps involved in solving this equation are squaring both sides, rearranging the equation, and using the quadratic formula. We hope that this article has provided a clear and step-by-step guide to solving such equations.

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

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In our previous article, we provided a step-by-step guide to solving quadratic equations involving square roots. In this article, we will answer some of the most frequently asked questions about quadratic equations.

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 can be written in the standard form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I solve a quadratic equation?

A: There are several methods to solve a quadratic equation, including factoring, using the quadratic formula, and completing the square. The method you choose depends on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that provides the solutions to a quadratic equation. It is given by:

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

Q: When do I use the quadratic formula?

A: You use the quadratic formula when the quadratic equation cannot be factored easily or when you need to find the solutions to a quadratic equation.

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

A: A linear equation is a polynomial equation of degree one, which means the highest power of the variable is one. A quadratic equation, on the other hand, is a polynomial equation of degree two.

Q: Can a quadratic equation have more than two solutions?

A: No, a quadratic equation can have at most two solutions.

Q: Can a quadratic equation have no real solutions?

A: Yes, a quadratic equation can have no real solutions. This occurs when the discriminant ($b^2 - 4ac$) is negative.

Q: How do I determine the number of solutions to a quadratic equation?

A: You can determine the number of solutions to a quadratic equation by looking at the discriminant. If the discriminant is positive, the equation has two real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: Can a quadratic equation be used to model real-world problems?

A: Yes, quadratic equations can be used to model a wide range of real-world problems, including projectile motion, optimization problems, and electrical circuits.

Q: What are some common applications of quadratic equations?

A: Some common applications of quadratic equations include:

• Projectile motion
• Optimization problems
• Electrical circuits
• Physics and engineering
• Computer graphics

Conclusion

In this article, we answered some of the most frequently asked questions about quadratic equations. We hope that this article has provided a useful resource for students and professionals alike. If you have any further questions or need further clarification, please don't hesitate to contact us.

Additional Resources

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

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

We hope that this article has provided a useful resource for students and professionals alike. If you have any questions or need further clarification, please don't hesitate to contact us.