Problem 1 - Algebra 2

Question

Solve each equation by using the Square Root Property. \(x^{2}+14 x+49=9\)

Step-by-Step Solution

Answer

The solutions are \(x = -4\) and \(x = -10\).

Step 1: Identify the Quadratic Equation in Standard Form

The equation given is \(x^2 + 14x + 49 = 9\). Before applying the square root property, we need to express it in standard form, which involves having zero on one side of the equation. This is already in a quadratic complete square form, except for the numeric adjustment.

Step 2: Rearrange the Equation to a Perfect Square Form

Observe that \(x^2 + 14x + 49\) is a perfect square trinomial. Recognize that it can be expressed as \((x + 7)^2 = 49\). So, adjust the equation to \((x + 7)^2 = 9+7^2 \), which remains as \((x+7)^2 = 9\).

Step 3: Apply the Square Root Property

Using the square root property, we take the square root of both sides of the equation: \(\sqrt{(x+7)^2} = \pm\sqrt{9}\), giving us \(x+7 = \pm3\).

Step 4: Solve the Linear Equations

Now, solve for \(x\) in two separate linear equations: \(x + 7 = 3\) and \(x + 7 = -3\).1. From \(x + 7 = 3\), subtract 7 from both sides to obtain \(x = 3 - 7 = -4\).2. From \(x + 7 = -3\), subtract 7 from both sides to obtain \(x = -3 - 7 = -10\).

Step 5: Conclude the Solution

The solutions to the original equation \((x + 7)^2 = 9\) are \(x = -4\) and \(x = -10\). Therefore, these are the values of \(x\) which satisfy the equation.

Key Concepts

Square Root Property

The Square Root Property is a handy method for solving quadratic equations, especially when they are already in the form of a perfect square. Basically, if you have an equation of the form \[(x+a)^2 = b\]You can solve it by taking the square root of both sides. This means the expression inside the square, \[x+a\],will equal to \[ \pm \sqrt{b}\].This gives us two linear equations to solve, one for the positive square root and another for the negative square root. For example, with the equation \[(x+7)^2=9\],you apply the square root property like so:

• Take the square root of both sides: \(\sqrt{(x+7)^2} = \pm\sqrt{9}\)
• The result is two possible solutions: \((x+7)=3\) or \((x+7)=-3\)

This step transforms the problem from a quadratic equation to two simpler linear equations, making it much easier to solve.

Completing the Square

Completing the Square is a method used to transform any quadratic equation into a perfect square trinomial, which makes solving the equation using the Square Root Property straightforward. A standard quadratic equation looks like:\[ax^2 + bx + c = 0\].The goal is to rewrite this in the form of \[(x+d)^2 = e\].To complete the square, follow these steps:

• Divide all terms by \(a\) (if \(a eq 1\)) to simplify the coefficient of \(x^2\).
• Move the constant term \(c\) to the other side of the equation.
• Take half of the coefficient of \(x\), square it, and add it to both sides of the equation.
• The left side of the equation is now a perfect square trinomial, like \((x+d)^2\).

Once you have a perfect square, you can then solve for \(x\) using the square root property. Understanding this concept not only helps in solving quadratic equations but also deepens your insight into the structure and properties of quadratic functions.

Perfect Square Trinomials

Perfect square trinomials are a special type of trinomial that can be expressed as \[(x+a)^2\].Recognizing them is crucial when solving quadratic equations, as it allows quick simplification. These trinomials have the general expanded form:\[x^2 + 2ax + a^2\].Looking at our example equation \(x^2 + 14x + 49\), this can be rewritten as the perfect square \[(x+7)^2\],since \(2(7)=14\) and \(7^2=49\).Identifying a perfect square trinomial involves checking:

• The middle term should be twice the product of the square root of the first and third terms.
• The third term should be the square of what you get in completing the square process.

When you can successfully express a trinomial as a perfect square, it simplifies the process of solving the equation using methods like the Square Root Property. Keep practicing the recognition and factoring of perfect square trinomials to improve your problem-solving skills in algebra.