Problem 14 - College Algebra and Calculus: An Applied Approach

Question

Use the Quadratic Formula to solve the quadratic equation. $$ x^{2}-10 x+22=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x = 5 + \sqrt{3} \) and \( x = 5 - \sqrt{3} \).

Step 1: Identify a, b, and c

In the given equation \( x^2 - 10x + 22 = 0 \), 'a' is the coefficient of the quadratic term (x^2), so \( a = 1 \). 'b' is the coefficient of the linear term (x), so \( b = -10 \). 'c' is the constant term, so \( c = 22 \).

Step 2: Substitute into the Quadratic Formula

Substitute B, A, and C into the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). This gives: \( x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4*1*22}}{2*1} \).

Step 3: Simplify the Expression

First simplify the expression under the square root (discriminant): \(-10^2 - 4*1*22 = 100 - 88 = 12\). Next, compute: \( x = \frac{10 \pm \sqrt{12}}{2} \).

Step 4: Solve for x

Finally, calculate the two possible values for x: \( x = 5 + \sqrt{3} \) and \( x = 5 - \sqrt{3} \).

Key Concepts

Solving Quadratic Equations

Understanding the process to solve quadratic equations is essential in algebra. A quadratic equation is represented as an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \) and \( c \) are coefficients, and \( a \eq 0 \). The solutions to these equations, also known as roots or zeros, can be found using several methods, including factoring, graphing, completing the square, and using the Quadratic Formula.

The Quadratic Formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), is a reliable method that will always find the roots provided that the equation has a solution. To use this formula, simply plug in the values of \( a \), \( b \) and \( c \) from the equation into the formula and calculate the result.

Discriminant

The discriminant is a key part of the Quadratic Formula, located under the square root sign: \( b^2 - 4ac \). It tells us the nature of the roots of a quadratic equation without actually solving the equation.

• If the discriminant is positive, there are two distinct real number solutions.
• If it is zero, there is exactly one real number solution, also known as a repeated or double root.
• If the discriminant is negative, there are two complex solutions, and no real solutions exist.

In our example, \( (-10)^2 - 4\cdot1\cdot22 = 100 - 88 = 12 \), the discriminant is positive indicating that two distinct real number solutions exist for the given equation.

Quadratic Term Coefficients

The coefficients in a quadratic equation \( ax^2 + bx + c = 0 \) have specific roles. The coefficient \( a \) is in front of the quadratic term (\( x^2 \)) and is crucial as it determines the parabola's opening direction and its width. A positive \( a \) means the parabola opens upwards, while a negative \( a \) indicates it opens downwards.

The coefficient \( b \) is in front of the linear term (\( x \)), and it affects the vertex's horizontal location on the graph. Lastly, the constant term \( c \) gives the y-intercept of the parabola. In the given example, \( a = 1 \) which suggests the parabola opens upward, \( b = -10 \) helps determine the axis of symmetry, and \( c = 22 \) is where the parabola intersects the y-axis.