91Ó°ÊÓ

Factoring is a method used to simplify algebraic expressions and equations by breaking them down into simpler components that multiply together to give the original expression. In the equation given, \[x^2 \cdot 10^x - x \cdot 10^x = 2 \cdot 10^x\],the common factor among the terms on the left-hand side is \(10^x\). By factoring out \(10^x\), we rewrite the equation as:\[10^x(x^2 - x) = 2 \cdot 10^x\].This step reduces the complexity of the equation and makes it simpler to solve.

When you factor, look for:

• Common numbers or variables across terms.
• Patterns like differences of squares or perfect squares.

Factoring helps reveal the internal structure of the equation, making it easier to work with.

The quadratic formula is a universal tool used to solve quadratic equations of the form \[ax^2 + bx + c = 0\].The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

To use the formula, follow these steps:

• Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Substitute these values into the formula.
• Calculate the discriminant, \(b^2 - 4ac\), and find its square root.
• Apply the '+/-' part to find the two possible solutions for \(x\).

This formula is powerful because it works even for complex solutions.In our example, with \(a = 1\), \(b = -1\), and \(c = -2\), substituting these values into the quadratic formula gives us\(x = \frac{1 \pm 3}{2}\),resulting in solutions \(x = 2\) and \(x = -1\).

The discriminant is a key part of the quadratic formula, represented by \(b^2 - 4ac\). It tells us the nature of the roots of a quadratic equation.

Depending on its value:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• If negative, the solutions are complex numbers.

In our problem, the discriminant is calculated as \[(-1)^2 - 4 \times 1 \times (-2) = 9\].A positive discriminant (9 in our case) indicates two distinct real solutions, confirming that \(x = 2\) and \(x = -1\) are indeed real solutions.

Solving equations, especially quadratic ones, involves finding values of the variable that make the equation true. The process often involves multiple steps: identifying the type of equation, simplifying it, and applying methods like factoring or quadratic formula as appropriate.

For the equation\[x^2 - x = 2\],we rearranged it into a standard quadratic form\[x^2 - x - 2 = 0\].Once in this form, you choose the best solving strategy: factoring, completing the square, or the quadratic formula.

Checking solutions is crucial. Substituting your results back into the original equation ensures correctness. Both solutions \(x = 2\) and \(x = -1\) checked out when re-inserted into the equation, confirming their validity. Each step in solving ensures thorough understanding and accuracy, making the process an essential part of learning mathematics.