Problem 2 Solve the following Type II quad... [FREE SOLUTION]

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Painless Algebra

Lynette Long

$Math Studyset 91影视 Explanations$ Math

2 Edition

Chapter 9: Problem 2

Solve the following Type II quadratic equations. $$ x^{2}+4 x=0 $$

Short Answer

Expert verified

The solutions are $ x = 0 $ and $ x = -4 $.

Step by step solution

- Factor the quadratic equation

Look for a common factor in the equation. In this case, both terms in the equation have an $x$ as a common factor. Factoring gives:\[ x(x + 4) = 0 \]

- Set each factor to zero

Set each factor equal to zero and solve for $x$. This provides the solutions to the equation.1. $ x = 0 $2. $ x + 4 = 0 $

- Solve each equation

Solve each equation from Step 2:1. $ x = 0 $ is already solved.2. For $ x + 4 = 0 $, subtract 4 from both sides to get:\[ x = -4 \]

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Key Concepts

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

Factoring Quadratics

In algebra, factoring quadratic equations is a fundamental technique for solving them. A quadratic equation typically looks like $ax^2 + bx + c = 0$. To factor it, we look for two expressions that multiply together to give the original quadratic. For the equation in our example $x^2 + 4x = 0$, notice both terms share an $x$ as a common factor.
So, we can factor out the $x$ to get:
\[ x(x+4) = 0 \].
At this point, the quadratic equation is expressed as the product of its factors. Factoring simplifies the equation into a form where we can use the Zero Product Property.

Zero Product Property

Once we have factored the quadratic equation, we can use the Zero Product Property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In other words, if $A \times B = 0$, then $A = 0$ or $B = 0$.
For our factored equation $x(x + 4) = 0$, we apply this property by setting each factor equal to zero.
This gives us two separate equations:

$x = 0$
$x + 4 = 0$

If either of these is true, the original equation is satisfied.

Algebra

Algebra involves solving equations and working with variables. This Type II quadratic equation illustrates several key algebraic concepts.
Firstly, recognizing and factoring common terms simplifies the equation.
Secondly, using properties like the Zero Product Property helps break down problems into simpler parts.
Finally, solving the equations produced by these methods provides the solutions to the original problem. For $x + 4 = 0 $, subtract 4 from both sides:
\[ x = -4 \].
Thus, the solutions to the original equation $x^2 + 4x = 0 $ are $x = 0 $ and $x = -4$.
Understanding these algebraic techniques allows you to handle more complex problems efficiently.

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91影视

Solve the following Type II quadratic equations. $$ x^{2}+4 x=0 $$

Short Answer

Step by step solution

- Factor the quadratic equation

- Set each factor to zero

- Solve each equation

Key Concepts

Factoring Quadratics

Zero Product Property

Algebra

One App. One Place for Learning.

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