Question

Solve the equation by completing the square. \(5 x(x+6)=-50\)

Short answer

The solutions to the equation are \(x = 0\) and \(x = -6\).

Step-by-step solution

Expand the Equation

The equation \(5x(x+6)=-50\) is expanded as \(5x^2+30x=-50\).

Rearrange the Equation

Move -50 to the other side of the equation to gather like terms: \(5x^2+30x+50=0\).

Divide by the Leading Coefficient

Next, simplify the equation by dividing every term by 5: \(x^2+6x+10=0\).

Complete the Square

Next, complete the square which can be realized as (x+3)^2 = 9. So, rewrite the equation as \((x+3)^2 = 9\).

Solve for x

Take the square root of both sides, remembering to consider both the positive and negative roots: \(x+3 = ± √9\). So, \(x= -3 ± 3\). Thus, x could be 0 or -6.

Key concepts

Quadratic Equations

Quadratic equations are an essential part of algebra, representing expressions where the highest power of the variable is square. Typically, a quadratic equation takes the standard form of \(ax^2 + bx + c = 0\). In these equations:

• \(a\) is the coefficient of \(x^2\)
• \(b\) is the coefficient of \(x\)
• \(c\) is the constant term

Solving quadratic equations means finding the value(s) of \(x\) for which the equation holds true. The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is one common method of solving these equations.
However, there are other techniques like factoring, using the quadratic formula, and completing the square. Completing the square is a method particularly useful because it allows for deriving the quadratic formula, also offering insights in graphing parabolas.

Solving Equations

Solving equations is a primary goal in algebra, aimed at finding the unknown variable. To solve an equation, particularly a quadratic one, we must rearrange and manipulate it into a solvable form. Let’s break it down:

• First, ensure the equation is fully expanded and all like terms are gathered on one side.
• Terms should often be rearranged so the equation aligns closely with its standard form.
• When solving, always check that each manipulation of the equation is valid and keeps the equation balanced.

In the example given, we solved the equation by completing the square. This involves forming a perfect square trinomial from the quadratic expression, which simplifies solving for \(x\). The equation changes structure to help find the solution more straightforwardly.
Remember, when solving any quadratic equation, check both possible solutions, especially when we take square roots, resulting in positive and negative roots.

Algebraic Manipulation

Algebraic manipulation refers to the process of treating an equation to isolate the variable. This requires a series of logical steps to simplify or restructure the equation, maintaining equality. In the problem like the one above:

• Begin by expanding any products or combined terms.
• Next, gather like terms to one side of the equation for simplification.
• Divide or multiply to remove coefficients from the variable you're solving for.

Algebraic manipulation relies heavily on basic arithmetic operations: addition, subtraction, multiplication, and division.
Completing the square is a type of algebraic manipulation where you adjust the quadratic to form a perfect square trinomial. This step helps in expressing the equation in a simple form such as \( (x + 3)^2 = 9\). It's crucial to understand how to "balance" these operations to maintain equality throughout the solving process, ensuring that the variable remains isolated and solutions are accurately derived.