Solve The Equation: ${ 4x^2 - 4x = -1 }$

Mar 12, 2025 by ADMIN 42 views

$**Solving Quadratic Equations: A Step-by-Step Guide to ${ 4x^2 - 4x = -1 }$**$

Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

In this article, we will focus on solving the quadratic equation ${ 4x^2 - 4x = -1 }$. This equation can be rewritten as $4x^2 - 4x + 1 = 0$ , which is a standard form of a quadratic equation. We will use various methods to solve this equation, including factoring, completing the square, and the quadratic formula.

Factoring the Quadratic Equation

One of the simplest methods to solve a quadratic equation is by factoring. Factoring involves expressing the quadratic equation as a product of two binomials. In this case, we can rewrite the equation as $(2x - 1)^2 = 0$ . This is a perfect square trinomial, which can be factored as a squared binomial.

To factor the equation, we need to find two binomials whose product is equal to the original equation. In this case, the binomials are $(2x - 1)$ and $(2x - 1)$ . When we multiply these binomials, we get $(2x - 1)^2 = 4x^2 - 4x + 1$ , which is equal to the original equation.

Solving for x

Now that we have factored the equation, we can solve for $x$ . To do this, we need to set each binomial equal to zero and solve for $x$ . In this case, we have:

$(2x - 1) = 0$

Solving for $x$ , we get:

$2x = 1$

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

This is the solution to the equation.

Completing the Square

Another method to solve a quadratic equation is by completing the square. Completing the square involves rewriting the quadratic equation in a form that allows us to easily solve for $x$ . In this case, we can rewrite the equation as $4x^2 - 4x + 1 = 0$ .

To complete the square, we need to add and subtract a constant term to the equation. In this case, we can add and subtract $\left(\frac{b}{2}\right)^2$ to the equation, where $b$ is the coefficient of the $x$ term. In this case, $b = -4$ , so we can add and subtract $\left(\frac{-4}{2}\right)^2 = 4$ to the equation.

Rewriting the Equation

After adding and subtracting 4 to the equation, we get:

$4x^2 - 4x + 1 + 4 = 0 + 4$

This can be rewritten as:

$4x^2 - 4x + 5 = 4$

Solving for x

Now that we have rewritten the equation, we can solve for $x$ . To do this, we need to isolate the $x$ term. In this case, we can subtract 4 from both sides of the equation to get:

$4x^2 - 4x + 1 = 0$

This is the same equation we started with. However, we can now use the quadratic formula to solve for $x$ .

The Quadratic Formula

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

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

In this case, $a = 4$ , $b = -4$ , and $c = 1$ . Plugging these values into the formula, we get:

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(4)(1)}}{2(4)}$

Simplifying the expression, we get:

$x = \frac{4 \pm \sqrt{16 - 16}}{8}$

$x = \frac{4 \pm \sqrt{0}}{8}$

$x = \frac{4}{8}$

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

This is the solution to the equation.

Conclusion

In this article, we have solved the quadratic equation ${ 4x^2 - 4x = -1 }$. We have used various methods to solve the equation, including factoring, completing the square, and the quadratic formula. We have found that the solution to the equation is $x = \frac{1}{2}$ .

Final Thoughts

Solving quadratic equations can be a challenging task, but with practice and patience, it can become second nature. By mastering the techniques of factoring, completing the square, and the quadratic formula, we can solve a wide range of quadratic equations and apply this knowledge to various fields.

In conclusion, solving quadratic equations is an essential skill that can be applied to various areas of mathematics and science. By understanding how to solve quadratic equations, we can make informed decisions and solve real-world problems.

References

[1] "Quadratic Equations" by Math Open Reference
[2] "Solving Quadratic Equations" by Khan Academy
[3] "Quadratic Formula" by Wolfram MathWorld

Additional Resources

[1] "Quadratic Equations" by MIT OpenCourseWare
[2] "Solving Quadratic Equations" by Purplemath
[3] "Quadratic Formula" by Mathway

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. In our previous article, we discussed how to solve quadratic equations using various methods such as factoring, completing the square, and the quadratic formula. In this article, we will answer some 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. The general form of a quadratic equation is $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

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

A: A quadratic equation can be factored if it can be expressed as a product of two binomials. To determine if a quadratic equation can be factored, we need to look for two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the $x$ term.

Q: What is the quadratic formula?

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

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

In this formula, $a$ , $b$ , and $c$ are the coefficients of the quadratic equation, and $x$ is the variable.

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

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

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

A: The quadratic formula and factoring are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while the quadratic formula involves using a formula to solve for $x$ .

Q: Can I use the quadratic formula to solve all quadratic equations?

A: Yes, the quadratic formula can be used to solve all quadratic equations. However, it may not always be the most efficient method, especially for equations that can be easily factored.

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, we need to look at the discriminant, which is the expression under the square root in the quadratic formula. 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 two complex solutions.

Q: Can I use the quadratic formula to solve quadratic equations with complex coefficients?

A: Yes, the quadratic formula can be used to solve quadratic equations with complex coefficients. However, we need to be careful when working with complex numbers, as they can be tricky to handle.

Q: How do I apply quadratic equations to real-world problems?

A: Quadratic equations have many real-world applications, such as physics, engineering, and economics. For example, we can use quadratic equations to model the motion of objects, the growth of populations, and the behavior of financial markets.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

Not checking if the equation can be factored before using the quadratic formula
Not simplifying the expression under the square root in the quadratic formula
Not checking if the discriminant is positive, zero, or negative before solving for $x$
Not being careful when working with complex numbers

Conclusion

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields such as physics, engineering, and economics. By understanding how to solve quadratic equations, we can apply this knowledge to real-world problems and make informed decisions. In this article, we have answered some frequently asked questions about quadratic equations and provided tips and tricks for solving them.