Question

Solve the equation by factoring. \(x^2-11 x=-30\)

Short answer

The solutions to the equation are \(x=6\) and \(x=5\).

Step-by-step solution

Arrange in Standard Form

To start solving the equation, the equation needs to be rearranged in the standard form: \(x^2-11x+30=0\). This is obtained by adding 30 to both sides.

Factor the Quadratic

The next step is to factor the quadratic equation. The factors of \(x^2-11x+30\) are \((x-6)\) and \((x-5)\), since -6 and -5 add up to -11 and multiply to 30. Hence, the factored form of the equation is \((x-6)(x-5)=0\).

Solve the Equations

The final step involves finding the roots of the quadratic equation by setting each factor equal to zero and solving for x. So, \(x-6=0\) gives \(x=6\), and \(x-5=0\) gives \(x=5\).

Key concepts

Standard Form

The standard form of a quadratic equation is vital for solving and understanding these equations. In essence, it presents the equation in the format:

• \(ax^2 + bx + c = 0\),

where \(a\), \(b\), and \(c\) are constants. Moving terms around to fit this format is often the first step in solving quadratic equations. In the given problem, we started with \(x^2 - 11x = -30\). To convert it into standard form, we added 30 to both sides, resulting in \(x^2 - 11x + 30 = 0\). This arrangement makes it clear which terms are which, and prepares the equation for further steps like factoring.

Roots of a Quadratic Equation

Understanding the roots of a quadratic equation is key to finding its solutions. The roots are simply the values of \(x\) that make the equation true, meaning that the equation equals zero. Once in standard form, you can solve for these roots through various methods, including factoring, using the quadratic formula, or graphing. In our example, after factoring the equation \((x-6)(x-5) = 0\), we found the roots by setting each factor equal to zero, resulting in \(x = 6\) and \(x = 5\). These roots are the solutions to the original equation, where both satisfy the equation \(x^2-11x+30=0\).

Factored Form

Factoring is a method used to simplify quadratic equations and find their roots. The factored form of a quadratic equation breaks it down into products of simpler expressions. Essentially, it looks like:

• \((px + q)(rx + s) = 0\),

where \(p, q, r,\) and \(s\) are numbers or expressions. In this example, we factored \(x^2 - 11x + 30\) into \((x - 6)(x - 5)\). This transformation is crucial because it turns the equation into a form where we can easily solve for the roots by setting each factor to zero.

Solving Equations

The ultimate goal of working with quadratic equations is often to solve them, finding values for \(x\) that make the equation true. Once an equation is in factored form, solving is straightforward. Set each factor equal to zero:

• If \((x - 6) = 0\), then \(x = 6\),
• and if \((x - 5) = 0\), then \(x = 5\).

These solutions are the points at which the graph of the quadratic equation intersects the x-axis. Understanding these steps provides a foundational skill in algebra that is applicable across many different types of math problems.