91Ó°ÊÓ

Factoring is a key method used to simplify and solve quadratic equations. In simple terms, factoring means expressing a quadratic equation as a product of its binomial factors. For the equation \(7x^2+28x+28=0\), the first step is to simplify it. We notice that 7 is a common factor in all the terms. Thus, we factor out 7, resulting in \(7(x^2+4x+4)=0\).
This simplification reveals an equation that looks like a perfect square trinomial. It can be expressed as \((x+2)^2\). Factoring is helpful because it breaks down complex equations into components that are easier to work with. Factoring turns the equation into \(7(x+2)^2=0\), ready to be further analyzed.

Once we factor the quadratic equation, we can solve for the variable \(x\). The key to solving factored equations is utilizing the zero-product property. This property states that if a product of terms equals zero, then at least one of the terms must be zero.
In our equation \(7(x+2)^2=0\), we focus on \((x+2)^2=0\). Since \((x+2)\) is repeated and 7 is a non-zero constant, our task is to solve for \(x+2=0\).
This step involves simple algebraic manipulation. We subtract 2 from both sides to isolate \(x\), resulting in \(x=-2\). This tells us that \(x=-2\) is our solution.
Solving an equation after factoring is a straightforward process if you apply the zero-product property effectively.

Verifying your solution is a crucial step to ensure you have solved the equation correctly. This process involves substituting the solution back into the original equation to check if the statement holds true.
For our solution \(x=-2\), substitute it into the original equation \(7x^2+28x+28=0\). This substitution gives us \(7(-2)^2 + 28(-2) + 28 = 0\). Calculating the terms, we have \(28 - 56 + 28 = 0\).
These calculations confirm that the left-hand side equals zero, just like the right-hand side. Therefore, \(x=-2\) is indeed a valid solution. Checking solutions guarantees the accuracy of your solution and bolsters your confidence in your problem-solving skills.