Solve The Equation:$4x^2 - 12x + 9 = 0$

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Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and economics. 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 equal to zero.

The Given Equation: $4x^2 - 12x + 9 = 0$

In this article, we will focus on solving the quadratic equation $4x^2 - 12x + 9 = 0$. This equation is a classic example of a quadratic equation, and it can be solved using various methods, including factoring, completing the square, and the quadratic formula.

Factoring the 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 factor the equation as follows:

$4x^2 - 12x + 9 = (2x - 3)^2 = 0$

Solving for x

To solve for x, we need to isolate the variable x. In this case, we can do this by taking the square root of both sides of the equation:

$2x - 3 = 0$

$x = \frac{3}{2}$

Completing the Square

Another method to solve a quadratic equation is by completing the square. This method involves manipulating the equation to express it in the form $(x + p)^2 = q$. In this case, we can complete the square as follows:

$4x^2 - 12x + 9 = 4(x^2 - 3x) + 9$

$= 4(x^2 - 3x + \frac{9}{4}) + 9 - 4(\frac{9}{4})$

$= 4(x - \frac{3}{2})^2 + 9 - 9$

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

Solving for x

To solve for x, we need to isolate the variable x. In this case, we can do this by taking the square root of both sides of the equation:

$4(x - \frac{3}{2})^2 = 0$

$x - \frac{3}{2} = 0$

$x = \frac{3}{2}$

The Quadratic Formula

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

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

In this case, we have $a = 4$, $b = -12$, and $c = 9$. Plugging these values into the quadratic formula, we get:

$x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(4)(9)}}{2(4)}$

$x = \frac{12 \pm \sqrt{144 - 144}}{8}$

$x = \frac{12 \pm \sqrt{0}}{8}$

$x = \frac{12}{8}$

$x = \frac{3}{2}$

Conclusion

In this article, we have solved the quadratic equation $4x^2 - 12x + 9 = 0$ using various methods, including factoring, completing the square, and the quadratic formula. We have shown that the solution to this equation is $x = \frac{3}{2}$. This solution can be verified by plugging it back into the original equation.

Final Answer

The final answer to the equation $4x^2 - 12x + 9 = 0$ is $x = \frac{3}{2}$.

Introduction

Quadratic equations are a fundamental concept in mathematics, and they play a crucial role in various fields, including physics, engineering, and economics. In our previous article, we solved the quadratic equation $4x^2 - 12x + 9 = 0$ using various methods, including 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 (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 equal to zero.

Q: How do I solve a quadratic equation?

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

Q: What is the quadratic formula?

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

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

Q: How do I use the quadratic formula?

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 and solve for x.

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

A: A quadratic equation is a polynomial equation of degree two, while a linear equation is a polynomial equation of degree one. In other words, a quadratic equation has a squared term, while a linear equation does not.

Q: Can I solve a quadratic equation by graphing?

A: Yes, you can solve a quadratic equation by graphing. By graphing the quadratic function, you can find the x-intercepts, which represent the solutions to the equation.

Q: What is the significance of the discriminant in a quadratic equation?

A: The discriminant is the expression under the square root in the quadratic formula. It can be used to determine the nature of the solutions to the equation. If the discriminant is positive, the equation has two distinct 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 I solve a quadratic equation with complex coefficients?

A: Yes, you can solve a quadratic equation with complex coefficients. The quadratic formula can be used to find the solutions to the equation, even if the coefficients are complex numbers.

Q: What is the relationship between quadratic equations and conic sections?

A: Quadratic equations are related to conic sections, which are curves that can be defined by a quadratic equation. Conic sections include circles, ellipses, parabolas, and hyperbolas.

Conclusion

In this article, we have answered some frequently asked questions about quadratic equations. We have covered topics such as the definition of a quadratic equation, methods for solving quadratic equations, and the significance of the discriminant. We hope that this article has been helpful in clarifying any questions you may have had about quadratic equations.

Final Answer

The final answer to the equation $4x^2 - 12x + 9 = 0$ is $x = \frac{3}{2}$.