Which Shows The Equation Below Written In The Form $a X^2 + B X + C = 0$?$x + 9 = 4(x - 1)^2$A. $4 X^2 - 7 X + 13 = 0$B. $4 X^2 - 9 X + 13 = 0$C. $4 X^2 - 7 X - 5 = 0$D. $4 X^2 - 9 X - 5 = 0$

Feb 27, 2025 by ADMIN 192 views

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

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 this article, we will explore how to solve quadratic equations in the form $ax^2 + bx + c = 0$ . We will also provide a step-by-step guide on how to rewrite a given equation in this form.

Understanding Quadratic Equations

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.

Rewriting the Equation

The given equation is $x + 9 = 4(x - 1)^2$ . To rewrite this equation in the form $ax^2 + bx + c = 0$ , we need to expand the right-hand side of the equation.

Expanding the Right-Hand Side

To expand the right-hand side of the equation, we need to use the binomial expansion formula. The binomial expansion formula is given by:

$(a + b)^n = \binom{n}{0} a^n b^0 + \binom{n}{1} a^{n-1} b^1 + \binom{n}{2} a^{n-2} b^2 + ... + \binom{n}{n-1} a^1 b^{n-1} + \binom{n}{n} a^0 b^n$

In this case, we have $(x - 1)^2$ , which can be expanded as:

$(x - 1)^2 = x^2 - 2x + 1$

Now, we can substitute this expression back into the original equation:

$x + 9 = 4(x^2 - 2x + 1)$

Distributing the 4

To distribute the 4, we need to multiply each term inside the parentheses by 4:

$x + 9 = 4x^2 - 8x + 4$

Rearranging the Terms

To rewrite the equation in the form $ax^2 + bx + c = 0$ , we need to rearrange the terms. We can do this by subtracting $x$ from both sides of the equation and subtracting 9 from both sides of the equation:

$0 = 4x^2 - 9x - 5$

Conclusion

In conclusion, the given equation $x + 9 = 4(x - 1)^2$ can be rewritten in the form $ax^2 + bx + c = 0$ as:

$0 = 4x^2 - 9x - 5$

Therefore, the correct answer is:

D. $4 x^2 - 9 x - 5 = 0$

Example Problems

Problem 1

Solve the equation $x + 2 = 3(x - 1)^2$ in the form $ax^2 + bx + c = 0$ .

Solution

To solve this problem, we need to follow the same steps as before. We can start by expanding the right-hand side of the equation:

$(x - 1)^2 = x^2 - 2x + 1$

Now, we can substitute this expression back into the original equation:

$x + 2 = 3(x^2 - 2x + 1)$

Distributing the 3, we get:

$x + 2 = 3x^2 - 6x + 3$

Rearranging the terms, we get:

$0 = 3x^2 - 7x + 1$

Therefore, the correct answer is:

A. $3 x^2 - 7 x + 1 = 0$

Problem 2

Solve the equation $x - 1 = 2(x + 2)^2$ in the form $ax^2 + bx + c = 0$ .

Solution

To solve this problem, we need to follow the same steps as before. We can start by expanding the right-hand side of the equation:

$(x + 2)^2 = x^2 + 4x + 4$

Now, we can substitute this expression back into the original equation:

$x - 1 = 2(x^2 + 4x + 4)$

Distributing the 2, we get:

$x - 1 = 2x^2 + 8x + 8$

Rearranging the terms, we get:

$0 = 2x^2 + 7x + 9$

Therefore, the correct answer is:

B. $2 x^2 + 7 x + 9 = 0$

Conclusion

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 this article, we will provide a Q&A guide on quadratic equations, covering topics such as solving quadratic equations, graphing quadratic functions, and more.

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 solve a quadratic equation?

A: To solve a quadratic equation, you can use the following steps:

Expand the right-hand side of the equation, if necessary.
Distribute the coefficients, if necessary.
Rearrange the terms to get the equation in the form $ax^2 + bx + c = 0$ .
Use the quadratic formula to find the solutions: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

Q: What is the quadratic formula?

A: The quadratic formula is a formula used to find the solutions of a quadratic equation. It is given by:

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

Q: How do I graph a quadratic function?

A: To graph a quadratic function, you can use the following steps:

Determine the vertex of the parabola, which is the point where the parabola changes direction.
Determine the x-intercepts of the parabola, which are the points where the parabola crosses the x-axis.
Plot the vertex and the x-intercepts on a coordinate plane.
Draw a smooth curve through the points to create the graph of the quadratic function.

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

A: A quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable. A quadratic function, on the other hand, is a function that can be written in the form $f(x) = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants, and $x$ is the variable.

Q: Can I use a calculator to solve a quadratic equation?

A: Yes, you can use a calculator to solve a quadratic equation. Most calculators have a built-in quadratic formula function that you can use to find the solutions of a quadratic equation.

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

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

Not expanding the right-hand side of the equation, if necessary.
Not distributing the coefficients, if necessary.
Not rearranging the terms to get the equation in the form $ax^2 + bx + c = 0$ .
Not using the quadratic formula correctly.

Q: Can I use quadratic equations in real-world applications?

A: Yes, quadratic equations have many real-world applications, including:

Physics: Quadratic equations are used to model the motion of objects under the influence of gravity.
Engineering: Quadratic equations are used to design and optimize systems, such as bridges and buildings.
Economics: Quadratic equations are used to model the behavior of economic systems, such as supply and demand.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and they have many real-world applications. By understanding how to solve quadratic equations, graph quadratic functions, and use the quadratic formula, you can apply quadratic equations to a wide range of problems and situations.