Problem 73 - Elementary Algebra

Question

(a) solve. (b) check. $$ z^{2}=11 z-18 $$

Step-by-Step Solution

Answer

The solutions are z = 9 and z = 2.

Step 1 Title - Rewrite the Equation

Rewrite the given quadratic equation in standard form: \[ z^2 - 11z + 18 = 0 \]

Step 2 Title - Identify Coefficients

Identify the coefficients for the quadratic equation \[az^2 + bz + c = 0 \]. In this case, \[ a = 1, b = -11, c = 18 \]

Step 3 Title - Use the Quadratic Formula

Use the quadratic formula \[ z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. Substituting the coefficients, we get: \[ z = \frac{-(-11) \pm \sqrt{(-11)^2 - 4(1)(18)}}{2(1)} \] Simplifying inside the square root: \[ z = \frac{11 \pm \sqrt{121 - 72}}{2} \] \[ z = \frac{11 \pm \sqrt{49}}{2} \]

Step 4 Title - Simplify the Roots

Solve for the roots by simplifying: \[ z = \frac{11 + 7}{2} \] and \[ z = \frac{11 - 7}{2} \] Therefore, \[ z = \frac{18}{2} = 9 \] and \[ z = \frac{4}{2} = 2 \]. So, the solutions are \[ z = 9 \] and \[ z = 2 \]

Step 5 Title - Check the Solutions

Substitute each solution back into the original equation to verify: For \[ z = 9 \]: \[ 9^2 = 11 \times 9 - 18 \] \[ 81 = 99 - 18 \] \[ 81 = 81 \] For \[ z = 2 \]: \[ 2^2 = 11 \times 2 - 18 \] \[ 4 = 22 - 18 \] \[ 4 = 4 \]. Both solutions are correct.

Key Concepts

quadratic formula

To solve quadratic equations, we often use the quadratic formula. It is a powerful tool to find the roots of any quadratic equation of the form \( ax^2 + bx + c = 0 \). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here’s how it works:

• The term \( -b \) flips the sign of \( b \).
• The \( \pm \) symbol means we need to calculate two values: one by adding the square root term and one by subtracting it.
• \( b^2 - 4ac \) is called the discriminant. It tells us about the nature of the roots:
• If it is positive, we get two real roots.
• If it is zero, there is exactly one real root.
• If it is negative, the roots are complex numbers.
• Finally, divide everything by \( 2a \) to get the roots.

This formula is crucial for finding the precise solutions to quadratic equations, making them easier to solve.

standard form

Before applying the quadratic formula, we need to rewrite the quadratic equation in its standard form. The standard form of a quadratic equation is: \[ ax^2 + bx + c = 0 \] Here’s what each part represents:

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

For the exercise problem \( z^2 = 11z - 18 \), we rearrange it to \[ z^2 - 11z + 18 = 0 \] Ensuring our equation is in standard form is a vital first step as it sets the stage for applying all consequent formulas and techniques accurately.

identifying coefficients

Identifying the coefficients is essential when we use the quadratic formula. These coefficients come from the standard form equation \[ ax^2 + bx + c = 0 \] For the equation in the exercise: \[ z^2 - 11z + 18 = 0 \], the coefficients are:

• \( a = 1 \), since the coefficient of \( z^2 \) is 1.
• \( b = -11 \), the coefficient of \( z \) is -11.
• \( c = 18 \), the constant term is 18.

With \( a, b, \) and \( c \) identified, we can substitute them directly into the quadratic formula to find the roots. This step ensures every calculation aligns with the actual equation's structure.

simplifying roots

After substituting the coefficients into the quadratic formula, we need to simplify the expression under the square root (the discriminant) and then the roots themselves. Let’s break it down using the discriminant: \[ b^2 - 4ac \]. For our coefficients \[ (a = 1, b = -11, c = 18) \], we calculate: \[ (-11)^2 - 4(1)(18) = 121 - 72 = 49 \] Thus, our discriminant is 49. We take its square root \[ \sqrt{49} = 7 \] Now, substitute back into the quadratic formula to get the roots: \[ z = \frac{11 \pm 7}{2} \] This gives us two solutions:

• \[ z = \frac{11 + 7}{2} = \frac{18}{2} = 9 \]
• \[ z = \frac{11 - 7}{2} = \frac{4}{2} = 2 \]

Hence, the roots are \( z = 9 \) and \( z = 2 \). Simplifying the roots helps ensure clarity and accuracy in our solutions, confirming that our calculations align correctly with the quadratic equation.