Problem 26 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Solve the equation by completing the square. $$ 7 p^{2}-20=0 $$

Step-by-Step Solution

Answer

The solution to the given equation \(7p^2 - 20 = 0\) by completing the square is \(p = \pm\sqrt{\frac{20}{7}}\).

Step 1: Write the given equation in standard form.

The given equation is \(7p^2 - 20 = 0\). We can rewrite it in standard form: $$ 7p^2 = 20 $$

Step 2: Divide the equation by the coefficient of \(p^2\).

The coefficient of the term \(p^2\) is 7. We need to get rid of this coefficient to create a perfect square trinomial. Let's divide both sides of the equation by 7: $$ p^2 = \frac{20}{7} $$

Step 3: Add and subtract a constant to create a perfect square trinomial.

To create a perfect square trinomial on the left-hand side of the equation, we need to add a constant to the equation such that the left side becomes \((p + a)^2\). To find this constant, we need to divide the coefficient of the linear term by 2 and square it: Since there is no linear term in our case, we will assume the coefficient to be 0: $$ a^2 = \left(\frac{0}{2}\right)^2 = 0 $$ This means that our equation is already a perfect square trinomial: $$ p^2 = \frac{20}{7} $$

Step 4: Unlock the square by taking the square root of both sides.

Now we can find the possible values of p by taking the square root of both sides of the equation: $$ \sqrt{p^2} = \pm\sqrt{\frac{20}{7}} $$ The positive and negative square roots of \(\frac{20}{7}\) are the solutions for p: $$ p = \pm\sqrt{\frac{20}{7}} $$ We have found the solution to the given equation by completing the square. The solutions for p are \(\pm\sqrt{\frac{20}{7}}\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are mathematical expressions where the highest power of the variable is squared. They typically have the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents the unknown variable. In the exercise we are discussing, the equation is slightly different, starting as \( 7p^2 - 20 = 0 \). Here, the term \( 7p^2 \) is quadratic, and our goal is to solve for the variable \( p \) that satisfies the equation.
Quadratic equations can have different types of solutions:

• Two distinct real solutions
• One unique real solution
• No real solutions but two complex ones

The types of solutions depend on the discriminant, calculated as \( b^2 - 4ac \). In this particular exercise, there is no linear term (meaning \( b = 0 \)), which simplifies the process significantly. The absence of the linear term allows us to solve it more easily by completing the square.

Mastering the Art of Solving Equations

Solving equations is a fundamental skill in algebra, crucial for finding the values of unknown variables that make the equation true. The equation given, \( 7p^2 - 20 = 0 \), is a classic example requiring us to explore the method of completing the square to find the solutions.
Solving by completing the square involves several systematic steps:

• Rewriting the equation in a form that allows completion of the square.
• Dividing the equation by a suitable factor to simplify it.
• Adding and subtracting necessary constants to form a perfect square trinomial.
• Taking the square root of both sides to solve for the variable.

In our exercise, the absence of a linear \( p \) term simplifies the process, leading directly to the solution. This method emphasizes rearranging and manipulating the equation strategically until we can easily derive the variable's values.

Exploring Mathematical Methods: Completing the Square

Completing the square is a powerful mathematical method used to solve quadratic equations, especially when factoring does not apply cleanly. This method simplifies solving by transforming a quadratic equation into a perfect square trinomial.
Here are steps to complete the square systematically:

• Ensure the quadratic term's coefficient is 1 for simplicity, accomplished by dividing the entire equation if necessary.
• Identify the constant that needs to be added to convert the expression into a perfect square by using the linear coefficient (even if it is zero).
• Add and subtract this constant appropriately to form an expression like \((x + a)^2\).
• Proceed by solving the equation after transforming it into a perfect square trinomial.

In the provided exercise, completing the square directly unravels the solution for \( p \) by only a few adjustments, ultimately leading to \( p = \pm\sqrt{\frac{20}{7}} \). This method provides a reliable and general approach to handling quadratics, offering an alternative to factoring and the quadratic formula. Each quadratic equation might present a slightly different challenge, but completing the square remains universally applicable.