Problem 28 Solve each equation. $$ -x^{... [FREE SOLUTION]

Chapter 5: Problem 28

Solve each equation. $$ -x^{2}-8 x=16 $$

Short Answer

Expert verified

x = -4

Step by step solution

Rewrite the Equation

First, rewrite the given equation in standard form. Add 16 to both sides to set the equation to zero: -x^{2} - 8x - 16 = 0

Factor the Equation

Next, factor the quadratic equation. The equation -x^{2} - 8x - 16 = 0 can be factored as follows: - (x+4)^{2} = 0

Solve for x

Solve for x by using the zero-product property. Since - (x+4)^{2} = 0 x + 4 = 0 x = -4

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Factoring Quadratic Equations

Factoring quadratic equations is a method used to break down a complex equation into simpler components. This method is especially useful for solving quadratic equations like the one in the exercise:

$ -x^{2} - 8x = 16 $. Start with rewriting it in standard form: $ -x^{2} - 8x - 16 = 0 $. Once in standard form, the next goal is to express it as a product of factors. For this particular equation, it factors to:
$ - (x+4)^{2} = 0 $. Factoring transforms a quadratic equation into a more manageable form, making it easier to solve for the variable.

Zero-Product Property

The zero-product property is a principle which states that if the product of two factors is zero, at least one of the factors must be zero. This is crucial for solving quadratic equations after factoring.

In the equation $ - (x+4)^{2} = 0 $, applying the zero-product property allows us to set the factor equal to zero:
$ x + 4 = 0 $. Since the equation simplifies to a single factor, solving $ x + 4 = 0 $ gives us the solution $ x = -4 $. This reduces the complexity of resolving quadratic equations by focusing on simpler, linear equations.

Standard Form of Quadratic Equations

Understanding the standard form of quadratic equations is vital for solving them. The standard form is expressed as $ ax^{2} + bx + c = 0 $, where $a, b,$ and $c$ are constants. This format aids in organizing the equation for easier manipulation.

For example, the given equation $ -x^{2} - 8x = 16 $ must first be converted to the standard form: $ -x^{2} - 8x - 16 = 0 $. This sets up the equation for factoring and the application of other solution methods. Always ensuring your quadratic equation is in standard form is a foundational step to solving these types of equations accurately.

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91影视

Solve each equation. $$ -x^{2}-8 x=16 $$

Short Answer

Step by step solution

Rewrite the Equation

Factor the Equation

Solve for x

Key Concepts

Factoring Quadratic Equations

Zero-Product Property

Standard Form of Quadratic Equations

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