Problem 58 - Elementary Algebra

Question

\(p^{2}-14 p+49=0\)

Step-by-Step Solution

Answer

p = 7

Step 1 - Identify the quadratic equation

The given equation is a quadratic equation of the form: ax^2 + bx + c = 0 where a = 1, b = -14 and c = 49.

Step 2 - Check for perfect square trinomial

Recognize that the given quadratic equation might be a perfect square trinomial. A perfect square trinomial has the form (x - d)^2 = 0 Compare it with the given equation: (p - 7)^2 = p^2 - 14p + 49 Here, d = 7 which fits perfectly.

Step 3 - Rewrite the equation as a squared term

Rewrite the given quadratic equation as a perfect square trinomial: (p - 7)^2 = 0

Step 4 - Solve for p

Now solve the equation for p by taking the square root of both sides: (p - 7)^2 = 0 p - 7 = 0 p = 7

Key Concepts

Perfect Square Trinomial

A perfect square trinomial is a special type of quadratic expression. It is formed by squaring a binomial. For example, \(a + b\)^2 or \(a - b\)^2.

This expression expands to: \( (a + b)^2 = a^2 + 2ab + b^2 \) or \( (a - b)^2 = a^2 - 2ab + b^2 \), respectively.

Identifying a perfect square trinomial in the quadratic equation \( p^2 - 14p + 49 = 0 \) lets us rewrite it as \( (p - 7)^2 \), which simplifies solving.

Quadratic Equation

A quadratic equation is any equation that can be written in the form: \ax^2 + bx + c = 0\

Here, 'a', 'b', and 'c' are constants, and 'x' is the variable.

In our specific problem, \( p^2 - 14p + 49 = 0 \), the coefficients are: a = 1, b = -14, and c = 49.

Recognizing that we are dealing with a quadratic equation is the first step towards solving it.

Roots of a Quadratic Equation

The roots of a quadratic equation are the values of 'x' that satisfy the equation \ax^2 + bx + c = 0\.

These roots can be found using various methods, such as factoring, completing the square, or using the quadratic formula: \x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

For our equation, we simplified it to \( (p - 7)^2 = 0 \).

Solving this, we find that \( p = 7 \) is the root.

Factoring Quadratics

Factoring is a process of breaking down an equation into simpler components that are easier to solve.

One method to factorize a quadratic equation like \( p^2 - 14p + 49 = 0 \) is to express it as \( (p - m)(p - n) = 0\.

Here, m and n are values that satisfy the equation \( m \times n = c \) and \ m + n = b \).

In our specific problem, we recognize it as a perfect square trinomial and factor it as \( (p - 7)^2 \). This makes solving the equation straightforward.