Problem 35 - Algebra A Combined Function

Question

Solve each equation by completing the square. See Examples 7 through 9 $$ x^{2}+8 x=-15 $$

Step-by-Step Solution

Answer

x = -4 ± √31

Step 1: Move Constant Term

Start by moving the constant term to the other side of the equation to make it easier to complete the square. The given equation is:\[ x^2 + 8x = -15 \]We will rewrite it as:\[ x^2 + 8x + \_ = 15 + \_ \] where we will fill in the blanks next.

Step 2: Calculate the Term to Complete the Square

To complete the square, take half of the coefficient of the x-term (which is 8), then square it.\[ \frac{8}{2} = 4 \]\[ 4^2 = 16 \]Add 16 to both sides of the equation. The equation becomes:\[ x^2 + 8x + 16 = 15 + 16 \]

Step 3: Rewrite as a Perfect Square

Rewrite the left side of the equation as a squared binomial:\[ (x + 4)^2 = 31 \]The equation is now written using a perfect square.

Step 4: Solve for x

Take the square root of both sides to solve for x: \[ x + 4 = \pm \sqrt{31} \]Now, solve for x by isolating it:\[ x = -4 \pm \sqrt{31} \]

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental part of algebra involving polynomials of the form: \[ ax^2 + bx + c = 0 \]where \( a \), \( b \), and \( c \) are constants. The most distinguishing feature of a quadratic equation is the presence of the term \( x^2 \), which signifies that the highest power of the variable \( x \) is 2.
To understand quadratic equations better, remember:

• The graph of a quadratic equation is a parabola.
• These parabolas can open upwards or downwards.
• Quadratic equations can have two real solutions, one real solution, or no real solutions.

The method of completing the square transforms the standard form into a perfect square trinomial, providing a straightforward way to solve these equations through symmetry. Instead of proceeding with the quadratic formula, completing the square offers a visual demonstration of why solutions arise, because the vertex form \( (x + h)^2 = k \) clearly shows the horizontal shift \( h \).
Solving quadratic equations by completing the square is not only an effective strategy, but also deepens your understanding of the nature and behavior of quadratics.

Techniques for Solving Equations

Equations can present challenges, but solving them effectively creates clarity in mathematics. For quadratic equations, there are several methods available:

• Factoring, when possible, is often the simplest approach.
• The quadratic formula provides solutions directly, even for complex situations.
• Completing the square offers an intuitive way to express equations as perfect squares.

Each method has its benefits and ideal scenarios for use. Completing the square, specifically, helps in understanding the underlying structure of the equation. It involves a few significant steps:

• Rewriting the equation by moving the constant term \( c \).
• Taking half of the coefficient of the linear term \( bx \), squaring it, and adding to both sides.
• Re-formulating the equation as a perfect square trinomial, making it look like \( (x + p)^2 \).

These steps not only solve the equation but also extend your algebraic skills, enabling you to recognize and utilize symmetry present in quadratic equations.

The Concept of Perfect Squares in Algebra

A perfect square is an expression formed by squaring a binomial. In algebra, the term "perfect square" refers to expressions like \( (x+a)^2 \). They emerge frequently in quadratic equations.
Completing the square is a specific technique used to rewrite a quadratic equation into a perfect square trinomial:

• To complete the square, take half of the coefficient of \( x \), square it, and add it to both sides of the equation.
• This practice turns the equation into \( x^2 + bx + c \) into \( (x + m)^2 = n \).

The benefit of rewriting an equation as a perfect square is that it simplifies the process of solving. It turns a complex quadratic into a format that's easily solvable with basic algebraic operations like taking square roots.
Understanding perfect squares also helps in geometry and in finding minimum or maximum values in functions, given that the vertex form describes transformations in the coordinate plane.