Solve For X X X . 4 X 2 + 24 X = − 35 4x^2 + 24x = -35 4 X 2 + 24 X = − 35

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Introduction

Solving quadratic equations is a fundamental concept in mathematics, and it is essential to understand how to solve them to progress in various mathematical disciplines. In this article, we will focus on solving a quadratic equation of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants. We will use the given equation $4x^2 + 24x = -35$ as an example to demonstrate the steps involved in solving a quadratic equation.

Understanding the Equation

The given equation is $4x^2 + 24x = -35$. To solve for $x$, we need to isolate the variable $x$ on one side of the equation. The first step is to move all the terms to one side of the equation by adding $35$ to both sides. This gives us:

$$4x^2 + 24x + 35 = 0$$

Rearranging the Equation

Now that we have the equation in the standard form $ax^2 + bx + c = 0$, we can proceed to solve for $x$. The next step is to factorize the equation, if possible. However, in this case, the equation does not factor easily, so we will use the quadratic formula to solve for $x$.

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It states that for an equation of the form $ax^2 + bx + c = 0$, the solutions for $x$ are given by:

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

In our case, $a = 4$, $b = 24$, and $c = 35$. Plugging these values into the quadratic formula, we get:

$$x = \frac{-24 \pm \sqrt{24^2 - 4(4)(35)}}{2(4)}$$

Simplifying the Expression

Now that we have the quadratic formula, we can simplify the expression under the square root. The expression inside the square root is $24^2 - 4(4)(35)$. Evaluating this expression, we get:

$$24^2 - 4(4)(35) = 576 - 560 = 16$$

So, the quadratic formula becomes:

$$x = \frac{-24 \pm \sqrt{16}}{8}$$

Solving for $x$

Now that we have the simplified expression, we can solve for $x$. The expression under the square root is $16$, which is a perfect square. Taking the square root of $16$, we get:

$$x = \frac{-24 \pm 4}{8}$$

Finding the Solutions

Now that we have the simplified expression, we can find the solutions for $x$. We have two possible solutions, one for the plus sign and one for the minus sign. Evaluating the expression for the plus sign, we get:

$$x = \frac{-24 + 4}{8} = \frac{-20}{8} = -\frac{5}{2}$$

Evaluating the expression for the minus sign, we get:

$$x = \frac{-24 - 4}{8} = \frac{-28}{8} = -\frac{7}{2}$$

Conclusion

In this article, we solved the quadratic equation $4x^2 + 24x = -35$ using the quadratic formula. We first rearranged the equation to the standard form $ax^2 + bx + c = 0$, and then used the quadratic formula to find the solutions for $x$. We found two possible solutions, $x = -\frac{5}{2}$ and $x = -\frac{7}{2}$.

Final Answer

The final answer is $\boxed{-\frac{5}{2}, -\frac{7}{2}}$.

Additional Resources

For more information on solving quadratic equations, please refer to the following resources:

• Quadratic Formula
• Solving Quadratic Equations
• Quadratic Equations

Related Topics

• Linear Equations
• Polynomial Equations
• Algebraic Equations

Tags

• Quadratic Equations
• Quadratic Formula
• Algebra
• Mathematics
• Solving Equations
  

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Introduction

In our previous article, we solved the quadratic equation $4x^2 + 24x = -35$ using the quadratic formula. In this article, we will answer some frequently asked questions related to solving quadratic equations.

Q&A

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 of the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: How do I know if a quadratic equation can be factored?

A: A quadratic equation can be factored if it can be written in the form $(x + p)(x + q) = 0$, where $p$ and $q$ are constants. To factor a quadratic equation, you need to find two numbers whose product is $ac$ and whose sum is $b$.

Q: What is the quadratic formula?

A: The quadratic formula is a formula for solving quadratic equations of the form $ax^2 + bx + c = 0$. 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 $c$ into the formula. Then, simplify the expression under the square root and solve for $x$.

Q: What is the difference between the quadratic formula and factoring?

A: The quadratic formula is a general method for solving quadratic equations, while factoring is a specific method that can be used to solve some quadratic equations. Factoring is a more efficient method when the equation can be factored easily, but the quadratic formula is a more general method that can be used to solve any quadratic equation.

Q: Can I use the quadratic formula to solve a quadratic equation with complex solutions?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex solutions. In this case, the expression under the square root will be negative, and you will need to use the imaginary unit $i$ to simplify the expression.

Q: How do I know if a quadratic equation has real or complex solutions?

A: To determine if a quadratic equation has real or complex solutions, you need to look at the expression under the square root. If the expression is positive, the equation has real solutions. If the expression is negative, the equation has complex solutions.

Q: Can I use the quadratic formula to solve a quadratic equation with a variable coefficient?

A: Yes, the quadratic formula can be used to solve quadratic equations with variable coefficients. In this case, you need to plug in the values of the variable coefficients into the formula and simplify the expression.

Conclusion

In this article, we answered some frequently asked questions related to solving quadratic equations. We covered topics such as the quadratic formula, factoring, and complex solutions. We hope that this article has been helpful in clarifying some of the concepts related to solving quadratic equations.

Final Answer

The final answer is $\boxed{0}$.

Additional Resources

For more information on solving quadratic equations, please refer to the following resources:

• Quadratic Formula
• Solving Quadratic Equations
• Quadratic Equations

Related Topics

• Linear Equations
• Polynomial Equations
• Algebraic Equations

Tags

• Quadratic Equations
• Quadratic Formula
• Algebra
• Mathematics
• Solving Equations