Problem 18 - Elementary Algebra

Question

\(p^{2}-10 p+9=0\)

Step-by-Step Solution

Answer

The solutions are \( p = 9 \) and \( p = 1 \).

Step 1: Identify the quadratic equation

Recognize that the given equation is a quadratic equation in the standard form: \[ p^2 - 10p + 9 = 0 \]

Step 2: Determine coefficients

Identify the coefficients for the quadratic equation in the form \(ax^2 + bx + c = 0\). Here: \[a = 1, \ b = -10, \ c = 9\]

Step 3: Apply the quadratic formula

The quadratic formula is given by: \[ p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substitute the identified coefficients into the quadratic formula: \[ p = \frac{-(-10) \pm \sqrt{(-10)^2 - 4(1)(9)}}{2(1)} \]

Step 4: Simplify inside the square root

Simplify the terms inside the square root: \[ p = \frac{10 \pm \sqrt{100 - 36}}{2} \]

Step 5: Evaluate the square root

Complete the square root calculation: \[ p = \frac{10 \pm \sqrt{64}}{2} \]. Since \( \sqrt{64} = 8 \): \[ p = \frac{10 \pm 8}{2} \]

Step 6: Find the possible solutions

Calculate the possible values for \( p \): \[ p = \frac{10 + 8}{2} = 9 \ \] and \[ p = \frac{10 - 8}{2} = 1 \]

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation. A quadratic equation has the form \[ ax^2 + bx + c = 0 \] where \( a \), \( b \), and \( c \) are coefficients. The quadratic formula states: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] This formula helps find the values of \( x \) that satisfy the equation. Simply plug in the values for \( a \), \( b \), and \( c \) from your equation. The ± symbol means you will get two solutions, one for addition and one for subtraction inside the expression under the square root. The quadratic formula is particularly useful when a quadratic equation doesn't factor easily. However, completing the square may also solve quadratic equations, but the quadratic formula is often the quickest method.

• The formula derives from the process of completing the square.
• It universally applies to all quadratic equations.

Standard Form

Quadratic equations must be in standard form to use the quadratic formula effectively. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \] In this format, \( a \), \( b \), and \( c \) represent numerical coefficients. The equation includes:

• \( ax^2 \): quadratic term
• \( bx \): linear term
• \( c \): constant term

Before applying the quadratic formula, ensure your equation looks like this by arranging all terms on one side and setting it equal to zero. Placing it in standard form also helps in identifying coefficients easily, which is a crucial step before applying any solving method, including the quadratic formula.

Coefficients

Coefficients in a quadratic equation are the numerical values obtained in its standard form \( ax^2 + bx + c = 0 \). Here:

• \( a \) is the coefficient of \( x^2 \) (the quadratic term)
• \( b \) is the coefficient of \( x \) (the linear term)
• \( c \) is the constant term

In the example equation \( p^2 - 10p + 9 = 0 \):

• \( a = 1 \)
• \( b = -10 \)
• \( c = 9 \)

Once you have identified these values, substitute them into the quadratic formula. Careful identification of these values is essential since any error might lead to incorrect solutions! Accurately identifying coefficients helps ensure you implement the formula correctly.

Simplifying Square Roots

When solving a quadratic equation, you often encounter a square root, as seen in the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] The expression under the square root, \( b^2 - 4ac \), is called the discriminant. Simplifying inside the square root is an essential step. Here's how:

• Calculate \( b^2 \)
• Calculate \( 4ac \)
• Subtract \( 4ac \) from \( b^2 \)

For instance: Given \( p^2 - 10p + 9 = 0 \), coefficients are: \( a = 1 \), \( b = -10 \), and \( c = 9 \). Substitute these into the square root term: \[ b^2 - 4ac = (-10)^2 - 4(1)(9) = 100 - 36 = 64 \] Then, simplify the square root: \( \sqrt{64} = 8 \). Once simplified, substitute back into the quadratic formula to find your solutions.