How do you factor x^2 - 5x?

Factoring the expression \(x^2 - 5x\) involves finding two numbers that multiply to give the constant term (which is 0 in this case) and add up to the coefficient of the linear term (which is -5). Let's break this down step by step.

Identifying the Expression

The expression we have is a quadratic in the form of \(ax^2 + bx + c\), where:

• a = 1 (the coefficient of \(x^2\))
• b = -5 (the coefficient of \(x\))
• c = 0 (the constant term)

Factoring Out the Common Term

In this case, we can see that both terms in the expression share a common factor of \(x\). Therefore, we can factor out \(x\) from the expression:

\(x^2 - 5x = x(x - 5)

Understanding the Result

Now, we have factored the expression into \(x(x - 5)\). This means that the original quadratic can be expressed as the product of \(x\) and \((x - 5)\). Each factor represents a potential solution to the equation \(x^2 - 5x = 0\).

Finding the Roots

If we set the factored expression equal to zero, we can find the values of \(x\) that satisfy the equation:

\(x(x - 5) = 0\)

This gives us two possible solutions:

• x = 0
• x - 5 = 0 → x = 5

Visualizing the Factors

To visualize this, think of the expression as representing a parabola that opens upwards. The roots we found, \(x = 0\) and \(x = 5\), are the points where the parabola intersects the x-axis. The factor \(x\) indicates that the parabola touches the x-axis at the origin, while the factor \((x - 5)\) shows that it crosses the x-axis at \(x = 5\).

Summary of the Process

In summary, to factor \(x^2 - 5x\), we:

• Identified the common factor, which was \(x\).
• Factored it out to get \(x(x - 5)\).
• Set the factored expression to zero to find the roots.

Thus, the complete factorization of \(x^2 - 5x\) is \(x(x - 5)\), and the solutions to the equation are \(x = 0\) and \(x = 5\).