Problem 52 - Elementary Algebra

Question

For the following problems, solve each of the quadratic equations using the method of extraction of roots. $$ (a+3)^{2}=49 $$

Step-by-Step Solution

Answer

Answer: The roots are \(a=4\) and \(a=-10\).

Step 1: Set up the equation

Write the given equation: \((a+3)^2=49\)

Step 2: Take the square root of both sides

To take the square root of both sides, we can write the equation as \(\sqrt{(a+3)^2}=\sqrt{49}\).

Step 3: Simplify the equation

Since the square root and square cancel each other, we are left with \(a+3=\pm\sqrt{49}\).

Step 4: Solve for a

Since \(\sqrt{49}=7\), the equation becomes \(a+3=\pm7\). We can now solve for \(a\) in two cases: Case 1: \(a+3=7\) Subtract 3 from both sides to get: \(a=4\) Case 2: \(a+3=-7\) Subtract 3 from both sides to get: \(a=-10\) So, the roots are \(a=4\) and \(a=-10\).

Key Concepts

Extraction of Roots

The extraction of roots is a method used to solve quadratic equations where the unknown variable is part of a square. The core idea behind this method is to isolate the squared term and then apply a square root to both sides of the equation to eliminate the exponent.

For instance, in the exercise \(a+3)^2=49\), the first step requires setting up the equation to highlight the square term. By applying the square root to both sides, the square is effectively 'extracted,' simplifying the term to \(a+3\) and removing the square term. This is a crucial part of solving quadratic equations; it reduces them to a simpler form that can readily be solved for the unknown variable.

It is important to remember that taking the square root of both sides of an equation introduces a plus and minus (\pm\) since both positive and negative numbers squared result in a positive outcome. Always account for both solutions when solving quadratic equations by this method.

Square Roots

Square roots are mathematical operations that answer the question: 'what number, when multiplied by itself, gives a certain value?' The square root of a number is denoted by the radical symbol \(\sqrt{}\).

For example, in the equation from the exercise \(a+3)^2 = 49\), taking the square root of both sides involves \(\sqrt{(a+3)^2} = \sqrt{49}\). The right side simplifies to \(7\) since \(7\) is the number that, when squared, equals \(49\). On the left side, the square root and the square cancel out, leaving us with the inside of the squared term: \(a+3\).

Understanding how the square root function interacts with squares is essential for solving quadratic equations. Also, remember that square roots can yield both positive and negative results (\pm\), which plays a pivotal role in determining all possible values of the variable being solved for.

Quadratic Formulas

The quadratic formula is a universal method for solving quadratic equations, commonly written in the standard form \(ax^2 + bx + c = 0\). The quadratic formula is \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). It uses the coefficients of the equation to find the roots or solutions for the variable.

This method is particularly useful when other methods, such as factorization or completing the square, are not applicable or yield complex solutions. The formula provides the exact roots by incorporating the operation of taking the square root of the discriminant (\(b^2 - 4ac\)), which determines the nature and number of roots.

The quadratic formula can seem intimidating, but with practice, it becomes an essential tool in a student's mathematical toolkit. It's directly linked to the concept of roots and is applicable for any quadratic equation, regardless of how simple or complex it is.