Problem 1 Solve the following Type II quad... [FREE SOLUTION]

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Painless Algebra

Lynette Long

$Math Studyset 91影视 Explanations$ Math

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Chapter 9: Problem 1

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

Short Answer

Expert verified

The solutions are $x = 0$ and $x = 2$.

Step by step solution

Write Equation in Standard Form

Ensure that the quadratic equation is in the form $ax^2 + bx + c = 0$. The given equation is already in this form as $x^2 - 2x = 0$.

Factor the Quadratic Equation

Factor the equation by finding common factors or using factorization methods. Here, $x^2 - 2x = 0$ can be factored as $x(x - 2) = 0$.

Solve for the Roots

Set each factor equal to zero and solve for $x$.$$\begin{aligned} x &= 0 \ x - 2 &= 0 \ x &= 2 \ \text{So, the solutions are} \ x &= 0 \text{ and } x = 2. \ \text{Therefore, the roots of the equation} \ x^2 - 2x = 0 \ \text{are} \ x = 0 \text{ and } x = 2. \text{ }$$

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Key Concepts

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

Standard Form

Writing a quadratic equation in standard form is the first vital step in solving it. The standard form of a quadratic equation is $ax^2 + bx + c = 0$. This structure helps in identifying the coefficients that will be useful in further steps such as factoring or applying the quadratic formula.

In our given problem, $x^2 - 2x = 0$, the equation is already in standard form. Here:

$a = 1$ (coefficient of $x^2$)
$b = -2$ (coefficient of $x$)
$c = 0$ (constant term)

Understanding and recognizing the standard form ensures that you can directly move on to the next step, whether it be factoring or using another method to solve the equation.

Factoring Quadratic Equations

Factoring is one effective method to solve quadratic equations. It involves expressing the equation as the product of its linear factors. For the quadratic equation $x^2 - 2x = 0$, one can notice a common factor in both terms: $x$.

We can rewrite the equation as: $x(x - 2) = 0$.
Factoring essentially reverses the process of distributing in algebra. In this case, you've identified that if $x(x - 2) = 0$ then $x$ and $x - 2$ must be the factors.
This method simplifies solving the equation since it breaks it down into simpler parts, allowing us to set each factor to zero in order to find the solutions.

Quadratic Roots

The roots or solutions of a quadratic equation are the values of $x$ that satisfy the equation $ax^2 + bx + c = 0$. After factoring the equation $x^2 - 2x = 0$ into $x(x - 2) = 0$, we can find the roots by setting each factor equal to zero:

$x = 0$
$x - 2 = 0$, therefore $x = 2$

These steps follow the zero-product property, which states that if a product of several factors is zero, at least one of the factors must be zero. Thus, the roots for the given equation are $x = 0$ and $x = 2$.

Identifying the roots is the final step in solving the quadratic equation, providing the specific values that satisfy the original equation $x^2 - 2x = 0$.

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91影视

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

Short Answer

Step by step solution

Write Equation in Standard Form

Factor the Quadratic Equation

Solve for the Roots

Key Concepts

Standard Form

Factoring Quadratic Equations

Quadratic Roots

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