Solve by Factoring -1.2(x+0.518)(x-1.718)=0
The problem presents an equation involving a quadratic expression in factored form and is set equal to zero. The equation features decimal coefficients and two binomials, each containing the variable x and a constant term, which are multiplied together and multiplied by a leading coefficient of -1.2. The task is to solve this equation for x, which presumably involves finding the values of x that make the entire expression equal to zero. To do so, one would typically apply the zero product property, which states that if the product of multiple factors equals zero, then at least one of the factors must itself be zero. This principle can be used to find the solutions for x by setting each binomial factor equal to zero and solving for x. However, since the instruction is to not answer the question, the detailed solution process is not provided.
$- 1.2 \left(\right. x + 0.518 \left.\right) \left(\right. x - 1.718 \left.\right) = 0$
Answer
Solution:
Step:1
To solve the equation $-1.2(x+0.518)(x-1.718)=0$, we recognize that if either factor $(x+0.518)$ or $(x-1.718)$ is zero, the product will be zero. Thus, we have two possibilities:
$$x + 0.518 = 0$$ $$x - 1.718 = 0$$
Step:2
We first address the factor $x + 0.518$ by setting it equal to zero to find the value of $x$.
Step:2.1
We write the equation:
$$x + 0.518 = 0$$
Step:2.2
To isolate $x$, we subtract $0.518$ from both sides:
$$x = -0.518$$
Step:3
Next, we consider the factor $x - 1.718$.
Step:3.1
We set this factor equal to zero:
$$x - 1.718 = 0$$
Step:3.2
We add $1.718$ to both sides to solve for $x$:
$$x = 1.718$$
Step:4
Combining the results from both factors, we find the solution set for the equation:
$$x = -0.518 , 1.718$$
Knowledge Notes:
To solve a quadratic equation by factoring, one must understand the Zero Product Property, which states that if the product of two factors is zero, then at least one of the factors must be zero. This property allows us to set each factor of a factored quadratic equation equal to zero and solve for the unknown variable.
The process of solving by factoring involves the following steps:
Factor the quadratic equation if it is not already factored.
Apply the Zero Product Property by setting each factor equal to zero.
Solve each resulting simple equation for the variable.
Combine all solutions to get the solution set for the original equation.
In the given problem, the equation is already factored, so we directly apply the Zero Product Property. The equation is in the form of $a(x+b)(x+c)=0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. We set each factor containing $x$ equal to zero and solve for $x$ to find the solutions.
