Problem 59 - Algebra A Combined Function

Question

Solve each equation. $$ x^{2}+14 x+49=0 $$

Step-by-Step Solution

Answer

The solution is \(x = -7\).

Step 1: Identify the Form of the Equation

The equation given is \(x^2 + 14x + 49 = 0\). This is a quadratic equation in the standard form \(ax^2 + bx + c = 0\), where \(a = 1\), \(b = 14\), and \(c = 49\).

Step 2: Recognize Perfect Square Form

Examine if the quadratic can be expressed as a perfect square. Notice that \(49\) is \(7^2\) and that \(14x\) is twice the product of \(x\) and \(7\). So the expression can be rewritten as \((x + 7)^2 = 0\).

Step 3: Solve the Perfect Square Equation

Since the equation is \((x + 7)^2 = 0\), set the expression inside the square to zero: \[x + 7 = 0\]

Step 4: Solve for x

Subtract 7 from both sides to solve for \(x\): \[x = -7\]

Key Concepts

Perfect Square

A perfect square is a number or expression that can be written as the square of another number or expression. It's essential in solving quadratic equations because it can simplify the process if the equation is a perfect square trinomial.
In the given exercise, the quadratic equation \(x^2 + 14x + 49 = 0\) is an example of a perfect square trinomial. To verify whether a quadratic is a perfect square trinomial, look for specific patterns.

• The first term \(x^2\) should be a square of \(x\).
• The last term \(49\) should be a perfect square, which it is because \(49 = 7^2\).
• The middle term \(14x\) should be twice the product of the number \(7\) and \(x\), which here is: \(2 \times 7 \times x = 14x\).

Recognizing such patterns allows us to rewrite the quadratic equation as \((x + 7)^2 = 0\). This step turns the equation into a more manageable form, where it becomes easier to solve.

Solving Equations

Solving equations is a fundamental aspect of algebra. When dealing with quadratic equations, there are several methods available, like factoring, completing the square, and using the quadratic formula. In this exercise, the equation is already in a form that invites the use of a perfect square, simplifying the solving process.
Once the equation \((x + 7)^2 = 0\) has been established, the next step in solving the equation is identifying the value of \(x\) that makes the equation true. Since the expression is squared and equals zero, the only solution is when the expression inside the parentheses itself equals zero.
Set \(x + 7 = 0\) and solve for \(x\):

• Subtract 7 from both sides of the equation.
• The solution is \(x = -7\).

Thus, the quadratic equation \(x^2 + 14x + 49 = 0\) has a single solution \(x = -7\), which is expected given it started as a perfect square.

Standard Form

The standard form of a quadratic equation is \(ax^2 + bx + c = 0\). This is a common format that helps in identifying the coefficients needed for solving equations by various methods. In our exercise \(x^2 + 14x + 49 = 0\), the coefficients are easy to identify: \(a = 1\), \(b = 14\), and \(c = 49\).
Recognizing the standard form in any quadratic equation is critical, as it allows the use of different solving techniques, like factoring or the quadratic formula. However, in this particular exercise, the equation’s structure led to the observation that it could be rewritten as a perfect square trinomial, facilitating a straightforward solution.
The benefit of identifying the standard form is that it establishes a consistent framework. Whether you aim to factor, complete the square, or use the quadratic formula, start by getting your equation into this form. It acts as a checklist for the properties and relationships between the terms, such as the sum and product of the roots and potential factorization opportunities.