Factoring the Leading Coefficient

Oct 13, 2025 by windailys.com 128 views

Understanding the Problem

In mathematics, the quadratic equation is a fundamental concept that is used to model various real-world situations. The standard form of a quadratic equation is given by $y = ax^2 + bx + c$ , where $a$ , $b$ , and $c$ are constants. In this article, we will focus on rewriting the quadratic equation $y=-2x^2+16x-1$ in vertex form using the method of completing the square.

Completing the Square: A Method for Rewriting Quadratic Equations

Completing the square is a technique used to rewrite a quadratic equation in vertex form. This method involves manipulating the quadratic equation to create a perfect square trinomial, which can then be factored into the square of a binomial. The vertex form of a quadratic equation is given by $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

Factoring the Leading Coefficient

To complete the square, Selma starts by factoring the leading coefficient from the first two terms of the quadratic equation. The leading coefficient is the coefficient of the $x^2$ term, which in this case is $-2$ . To factor the leading coefficient, Selma will divide the first two terms of the quadratic equation by $-2$ .

y = -2x^2 + 16x - 1

Dividing the First Two Terms by -2

To factor the leading coefficient, Selma will divide the first two terms of the quadratic equation by $-2$ . This will give her the expression:

y = -2(x^2 - 8x) - 1

The New Expression After Factoring the Leading Coefficient

After factoring the leading coefficient, Selma should have the following expression:

y = -2(x^2 - 8x) - 1

This is the new expression that Selma should have after factoring the leading coefficient. In the next step, we will discuss how to complete the square by adding and subtracting a constant term.

Completing the Square: Adding and Subtracting a Constant Term

To complete the square, Selma will add and subtract a constant term inside the parentheses. The constant term that needs to be added and subtracted is the square of half the coefficient of the $x$ term. In this case, the coefficient of the $x$ term is $-8$ , so half of this coefficient is $-4$ . The square of $-4$ is $16$ .

y = -2(x^2 - 8x + 16 - 16) - 1

Simplifying the Expression

After adding and subtracting the constant term, Selma can simplify the expression by combining like terms. This will give her the expression:

y = -2(x^2 - 8x + 16) + 32 - 1

The Final Expression After Completing the Square

After completing the square, Selma should have the following expression:

y = -2(x - 4)^2 + 31

This is the final expression that Selma should have after completing the square. In the next section, we will discuss the importance of completing the square and how it can be used to solve quadratic equations.

The Importance of Completing the Square

Completing the square is an important technique in mathematics that can be used to rewrite quadratic equations in vertex form. This method involves manipulating the quadratic equation to create a perfect square trinomial, which can then be factored into the square of a binomial. The vertex form of a quadratic equation is given by $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

Solving Quadratic Equations Using Completing the Square

Completing the square can be used to solve quadratic equations by rewriting them in vertex form. The vertex form of a quadratic equation can be used to find the vertex of the parabola, which can then be used to solve the equation. This method is particularly useful when the quadratic equation has a complex solution.

Conclusion

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic equation in vertex form. This method involves manipulating the quadratic equation to create a perfect square trinomial, which can then be factored into the square of a binomial.

Q: Why is completing the square important?

A: Completing the square is an important technique in mathematics that can be used to rewrite quadratic equations in vertex form. This method is particularly useful when the quadratic equation has a complex solution.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

Factor the leading coefficient from the first two terms of the quadratic equation.
Add and subtract a constant term inside the parentheses.
Simplify the expression by combining like terms.

Q: What is the vertex form of a quadratic equation?

A: The vertex form of a quadratic equation is given by $y = a(x - h)^2 + k$ , where $(h, k)$ is the vertex of the parabola.

Q: How do I find the vertex of a parabola?

A: To find the vertex of a parabola, you need to rewrite the quadratic equation in vertex form. The vertex of the parabola is given by the point $(h, k)$ in the vertex form of the equation.

Q: Can I use completing the square to solve quadratic equations?

A: Yes, you can use completing the square to solve quadratic equations. By rewriting the quadratic equation in vertex form, you can find the vertex of the parabola, which can then be used to solve the equation.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

Not factoring the leading coefficient from the first two terms of the quadratic equation.
Not adding and subtracting the correct constant term inside the parentheses.
Not simplifying the expression by combining like terms.

Q: Can I use completing the square to solve quadratic equations with complex solutions?

A: Yes, you can use completing the square to solve quadratic equations with complex solutions. By rewriting the quadratic equation in vertex form, you can find the vertex of the parabola, which can then be used to solve the equation.

Q: How do I know if I have completed the square correctly?

A: To check if you have completed the square correctly, you can plug the expression back into the original quadratic equation and simplify. If the expression is equal to the original quadratic equation, then you have completed the square correctly.

Q: Can I use completing the square to solve quadratic equations with rational coefficients?

A: Yes, you can use completing the square to solve quadratic equations with rational coefficients. By rewriting the quadratic equation in vertex form, you can find the vertex of the parabola, which can then be used to solve the equation.

Q: What are some real-world applications of completing the square?

A: Some real-world applications of completing the square include:

Modeling population growth and decline.
Analyzing the motion of objects under the influence of gravity.
Solving problems in physics and engineering.

Conclusion

In this article, we discussed some frequently asked questions about completing the square. We covered topics such as the importance of completing the square, how to complete the square, and some common mistakes to avoid. We also discussed some real-world applications of completing the square. By understanding these concepts, you can use completing the square to solve quadratic equations and model real-world problems.