Which Of The Following Values Of $x$ Satisfy The Equation $2x^2 + 7x + 3 = 0$?A. $x = \frac{1}{2}, X = 3$ B. $x = \frac{3}{2}, X = 1$ C. $x = -\frac{3}{2}, X = -1$ D. $x = -\frac{1}{2}, X = -3$

**Solving Quadratic Equations: A Step-by-Step Guide** =====================================================

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 explore the process of solving quadratic equations, using the equation $2x^2 + 7x + 3 = 0$ as a case study.

What is a Quadratic Equation?

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable (in this case, $x$) is two. The general form of a quadratic equation is:

$$ax^2 + bx + c = 0$$

where $a$, $b$, and $c$ are constants, and $a$ cannot be zero.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

This formula will give us two solutions for the equation, which may be real or complex numbers.

Solving the Equation $2x^2 + 7x + 3 = 0$

Using the quadratic formula, we can plug in the values of $a$, $b$, and $c$ from the equation $2x^2 + 7x + 3 = 0$:

$$a = 2, b = 7, c = 3$$

Substituting these values into the quadratic formula, we get:

$$x = \frac{-7 \pm \sqrt{7^2 - 4(2)(3)}}{2(2)}$$

Simplifying the expression under the square root, we get:

$$x = \frac{-7 \pm \sqrt{49 - 24}}{4}$$

$$x = \frac{-7 \pm \sqrt{25}}{4}$$

$$x = \frac{-7 \pm 5}{4}$$

This gives us two possible solutions for the equation:

$$x = \frac{-7 + 5}{4} = \frac{-2}{4} = -\frac{1}{2}$$

$$x = \frac{-7 - 5}{4} = \frac{-12}{4} = -3$$

Checking the Solutions

To verify that these solutions are correct, we can plug them back into the original equation:

For $x = -\frac{1}{2}$:

$$2\left(-\frac{1}{2}\right)^2 + 7\left(-\frac{1}{2}\right) + 3 = 2\left(\frac{1}{4}\right) - \frac{7}{2} + 3 = \frac{1}{2} - \frac{7}{2} + 3 = -3 + 3 = 0$$

For $x = -3$:

$$2(-3)^2 + 7(-3) + 3 = 2(9) - 21 + 3 = 18 - 21 + 3 = 0$$

Both solutions satisfy the equation, so we can be confident that they are correct.

Conclusion

In this article, we have explored the process of solving quadratic equations using the quadratic formula. We have used the equation $2x^2 + 7x + 3 = 0$ as a case study, and have shown that the solutions to the equation are $x = -\frac{1}{2}$ and $x = -3$. We have also verified that these solutions are correct by plugging them back into the original equation.

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 (in this case, $x$) is two.

Q: What is the quadratic formula?

A: The quadratic formula is a powerful tool for solving quadratic equations. It is given by:

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

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula, you need to plug in the values of $a$, $b$, and $c$ from the equation into the formula. Then, simplify the expression under the square root, and solve for $x$.

Q: What are the solutions to the equation $2x^2 + 7x + 3 = 0$?

A: The solutions to the equation $2x^2 + 7x + 3 = 0$ are $x = -\frac{1}{2}$ and $x = -3$.

Q: How do I verify that the solutions to a quadratic equation are correct?

A: To verify that the solutions to a quadratic equation are correct, you need to plug them back into the original equation and check if they satisfy the equation.