Find The Solution(s) To $2x^2 + 5x - 3 = 0$. Check All That Apply.A. $x = 3$ B. \$x = -3$[/tex\] C. $x = \frac{1}{2}$ D. $x = 2$ E. $x = -\frac{1}{2}$

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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 the quadratic equation $2x^2 + 5x - 3 = 0$, and we will check all the given solutions to see which ones are correct.

Understanding Quadratic Equations

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. In our equation, $2x^2 + 5x - 3 = 0$, a = 2, b = 5, and c = -3.

The Quadratic Formula

To solve a quadratic equation, we can use the quadratic formula, which is:

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

This formula will give us two solutions for the equation, which are the values of x that satisfy the equation.

Applying the Quadratic Formula

Now, let's apply the quadratic formula to our equation $2x^2 + 5x - 3 = 0$. We have a = 2, b = 5, and c = -3. Plugging these values into the quadratic formula, we get:

x=−5±52−4(2)(−3)2(2)x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)}

Simplifying the expression under the square root, we get:

x=−5±25+244x = \frac{-5 \pm \sqrt{25 + 24}}{4}

x=−5±494x = \frac{-5 \pm \sqrt{49}}{4}

x=−5±74x = \frac{-5 \pm 7}{4}

Solving for x

Now, we have two possible solutions for x:

x=−5+74x = \frac{-5 + 7}{4}

x=24x = \frac{2}{4}

x=12x = \frac{1}{2}

and

x=−5−74x = \frac{-5 - 7}{4}

x=−124x = \frac{-12}{4}

x=−3x = -3

Checking the Solutions

Now that we have found the solutions to the equation, let's check which ones are correct. We are given the following options:

A. $x = 3$ B. $x = -3$ C. $x = \frac{1}{2}$ D. $x = 2$ E. $x = -\frac{1}{2}$

From our calculations, we can see that the correct solutions are:

  • x = \frac{1}{2}$ (option C)

  • x = -3$ (option B)

Conclusion

In this article, we solved the quadratic equation $2x^2 + 5x - 3 = 0$ using the quadratic formula. We found two solutions for the equation, which are $x = \frac{1}{2}$ and $x = -3$. We also checked the given options and found that only options B and C are correct.

Final Answer

The final answer is:

  • x = \frac{1}{2}$ (option C)

  • x = -3$ (option B)

Note: The final answer is not a single number, but rather two values that satisfy the equation.

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them can be a challenging task for many students and professionals. In our previous article, we solved the quadratic equation $2x^2 + 5x - 3 = 0$ using the quadratic formula. In this article, we will answer some frequently asked questions about quadratic equations and provide additional information to help you better understand this topic.

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. The general form of a quadratic equation is $ax^2 + bx + c = 0$, 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: If the equation can be factored into the product of two binomials, we can solve it by setting each binomial equal to zero.
  • Quadratic formula: The quadratic formula is a general method for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
  • Graphing: We can also solve a quadratic equation by graphing the related function and finding the x-intercepts.

Q: What is the quadratic formula?

A: The quadratic formula is a general method for solving quadratic equations. It is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where a, b, and c are the coefficients of the quadratic equation.

Q: How do I apply the quadratic formula?

A: To apply the quadratic formula, we need to plug in the values of a, b, and c into the formula. We then simplify the expression under the square root and solve for x.

Q: What are the steps to solve a quadratic equation using the quadratic formula?

A: The steps to solve a quadratic equation using the quadratic formula are:

  1. Plug in the values of a, b, and c into the quadratic formula.
  2. Simplify the expression under the square root.
  3. Solve for x.

Q: What are the possible solutions to a quadratic equation?

A: The possible solutions to a quadratic equation are the values of x that satisfy the equation. These solutions can be real or complex numbers.

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

A: The number of solutions to a quadratic equation can be determined by the discriminant, which is the expression under the square root in the quadratic formula. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Q: What is the discriminant?

A: The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$.

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

A: To use the discriminant to determine the number of solutions to a quadratic equation, we need to plug in the values of a, b, and c into the discriminant and simplify the expression. If the discriminant is positive, the equation has two distinct solutions. If the discriminant is zero, the equation has one repeated solution. If the discriminant is negative, the equation has no real solutions.

Conclusion

In this article, we answered some frequently asked questions about quadratic equations and provided additional information to help you better understand this topic. We hope that this article has been helpful in clarifying any doubts you may have had about quadratic equations.

Final Answer

The final answer is:

  • The quadratic formula is a general method for solving quadratic equations.
  • The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
  • The discriminant is the expression under the square root in the quadratic formula, which is $b^2 - 4ac$.
  • The number of solutions to a quadratic equation can be determined by the discriminant.