Problem 32 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ x^{2}-14 x+45=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x = 9 \) and \( x = 5 \).

Step 1: Identify the coefficients

The given quadratic equation is \( x^2 - 14x + 45 = 0 \). Identify the coefficients: \( a = 1 \), \( b = -14 \), and \( c = 45 \).

Step 2: Calculate the discriminant

The discriminant \( \,\Delta\, \) is calculated using the formula \( b^2 - 4ac \). Substitute the values: \((-14)^2 - 4(1)(45) = 196 - 180 = 16\).

Step 3: Determine the nature of roots

Since the discriminant \( \,\Delta\, \) is positive (16), the quadratic equation has two distinct real roots.

Step 4: Apply the quadratic formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Plug in \( a = 1 \), \( b = -14 \), and \( c = 45 \): \( x = \frac{-(-14) \pm \sqrt{16}}{2(1)} \) which simplifies to \( x = \frac{14 \pm 4}{2} \).

Step 5: Calculate the roots

Substitute in the quadratic formula: \( x_1 = \frac{14 + 4}{2} = 9 \) and \( x_2 = \frac{14 - 4}{2} = 5 \).

Step 6: Verify the solutions

Substitute the solutions back into the original equation: For \( x = 9 \), \( 9^2 - 14(9) + 45 = 0 \). For \( x = 5 \), \( 5^2 - 14(5) + 45 = 0 \). Both verify the correctness of the solutions.

Key Concepts

Understanding the Discriminant

When dealing with quadratic equations, the discriminant is a handy tool to determine the nature of the roots of the equation. For a standard quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant \( \Delta \) is calculated using:

• \( \Delta = b^2 - 4ac \)

In this calculation:

• \( b \) is the coefficient of the linear term,
• \( a \) is the coefficient of the quadratic term,
• \( c \) is the constant term.

Depending on the value of \( \Delta \), you'll know the type of roots:

• If \( \Delta > 0 \), there are two distinct real roots.
• If \( \Delta = 0 \), there is exactly one real root, also known as a double root.
• If \( \Delta < 0 \), there are no real roots; instead, there are two complex roots.

In the given exercise, \( b^2 - 4ac \) was calculated as 16, which is greater than zero, indicating two distinct real roots.

Utilizing the Quadratic Formula

The quadratic formula is a powerful solution method for finding the roots of a quadratic equation. It is applicable for any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula itself is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

This formula provides the values of \( x \) that satisfy the equation:

• The symbol \( \pm \) indicates there will be two possible solutions, one for plus and one for minus.

Using the quadratic formula requires us to first find the discriminant \( \Delta = b^2 - 4ac \). Once the discriminant is known, you insert the values of \( a \), \( b \), and \( c \) into the formula to solve for \( x \). In the exercise, the quadratic formula was used successfully to find the roots \( x = 9 \) and \( x = 5 \). This method always guarantees a solution for any quadratic equation.

Identifying Real Roots

Real roots refer to solutions of the quadratic equation that are real numbers. These are the values of \( x \) that make the original equation true. Identifying whether the roots are real, complex, or a double root hinges on the discriminant calculation.
For our given exercise, once the discriminant \( \Delta = 16 \) was computed, it was concluded that the equation has two real roots. These real roots occur because \( \Delta > 0 \), resulting in:

• Two separate, intersecting points on the graph of the equation with the x-axis.

Thus, the solutions or real roots to the quadratic equation \( x^2 - 14x + 45 = 0 \) were found to be \( x = 9 \) and \( x = 5 \). It is essential to check these solutions by substituting them back into the original equation to confirm their validity, which was verified in the solution.