Problem 16 - Algebra 2

Question

Solve each equation by using the Square Root Property. \(x^{2}+4 x+4=25\)

Step-by-Step Solution

Answer

The solutions are \(x = 3\) and \(x = -7\).

Step 1: Rewrite the Equation in Standard Form

The equation given is \(x^{2}+4x+4=25\). First, we want to get all terms on one side to form a standard quadratic equation. Subtract 25 from both sides to get:\[ x^2 + 4x + 4 - 25 = 0 \]This simplifies to:\[ x^2 + 4x - 21 = 0 \].

Step 2: Review the Square Property

The Square Root Property states that if \(x^2 = k\), then \(x = \pm \sqrt{k}\). To use this property here, we first need to make the equation look like one side is a perfect square.

Step 3: Complete the Square

To complete the square, first take the middle term \(4x\), divide the coefficient of \(x\) by 2, and square it: \[ \left(\frac{4}{2}\right)^2 = 2^2 = 4 \].Add and subtract 4 inside the equation to help form a perfect square:\[ x^2 + 4x + 4 = 21 \].Know that \(x^2 + 4x + 4\) is \((x+2)^2\). Replace it in the equation:\[ (x+2)^2 = 25 \].

Step 4: Apply the Square Root Property

Apply the square root property to the equation:\[ \sqrt{(x+2)^2} = \pm \sqrt{25} \].This gives us:\[ x+2 = \pm 5 \].

Step 5: Solve for x

Solve the two equations created from taking the square root:1. \(x + 2 = 5\) Subtract 2 from both sides: \(x = 3\).2. \(x + 2 = -5\) Subtract 2 from both sides: \(x = -7\).Thus, the solutions are \(x = 3\) and \(x = -7\).

Key Concepts

Quadratic Equation

A quadratic equation is a fundamental algebraic equation that takes the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants with \( a eq 0 \). The equation is named 'quadratic' because the term "quad" refers to the square—that is, it deals with the square of a variable \( x^2 \). Quadratic equations appear in various fields such as physics, engineering, finance, and anywhere the relationship between variables is represented by a parabola.

Key characteristics of a quadratic equation include:

• The highest degree of the variable is 2.
• Its graph is a parabola, which can open upwards or downwards depending on the sign of \( a \).
• It may have two, one, or no real solutions.

In a quadratic equation like \( x^2 + 4x - 21 = 0 \), we identify \( a = 1 \), \( b = 4 \), and \( c = -21 \). To solve quadratic equations, various methods like factoring, using the quadratic formula, or completing the square are applied.

Understanding these basics and how coefficients affect the solutions will help to approach solving these equations effectively.

Complete the Square

Completing the square is a method used to solve quadratic equations. The aim is to transform a quadratic expression into a perfect square trinomial, making it easier to solve. This method is particularly useful when the quadratic doesn't factor neatly. Here's how it works:

• Begin with the standard form of the quadratic equation \( ax^2 + bx + c = 0 \).
• If \( a eq 1 \), divide every term by \( a \) to make the equation easier to manage.
• Take the linear coefficient \( b \), divide it by 2, and square the result. This value will complete the square.
• Add and subtract this squared value within the equation to maintain balance and help reformulate it into a perfect square form.

For example, in the expression \( x^2 + 4x \), we would take \((\frac{4}{2})^2 = 4 \). Adding and subtracting 4, we get \( x^2 + 4x + 4 \), which factors neatly into \((x+2)^2 \). The equation \( x^2 + 4x + 4 - 21 = 0 \) transforms first into \( (x+2)^2 = 25 \).

Completing the square is a powerful technique to incorporate when other methods like factoring or the quadratic formula are not straightforward or feasible.

Perfect Square

A perfect square is a quadratic expression that can be expressed in the form \((x + p)^2\), where \(p\) is any real number. Recognizing and working with perfect squares is a crucial skill in algebra. When we have a perfect square, solving for \( x \) becomes significantly easier, especially with the Square Root Property.

Knowing that \((x+2)^2\) is a perfect square in our given equation simplifies the process. Here’s why it is beneficial:

• It reduces complex quadratic expressions into simpler equations that are more manageable.
• Applying the Square Root Property becomes direct and effective since it states \( (x+p)^2 \) can be equated to \( (x+p) = \pm \text{the square root of the constant} \).
• This reduction leads to two straightforward equations to solve for \( x \), thus streamlining the steps to find the solution.

In the example \((x+2)^2 = 25\), it translates to \(x+2 = \pm 5\), immediately giving the solutions \( x = 3 \) and \( x = -7 \) after simple calculations. This systematic method is what makes handling perfect squares practical in solving quadratic equations.