Problem 16 - Algebra for College Students

Question

Use the quadratic formula to solve each of the quadratic equations. Check your solutions by using the sum and product relationships. $$a^{2}-6 a=2$$

Step-by-Step Solution

Answer

The solutions are \(a = 3 \pm \sqrt{11}\).

Step 1: Rewrite the equation in standard form

The quadratic formula is used on equations in the standard form of \( ax^2 + bx + c = 0 \). Start with the given equation \( a^2 - 6a = 2 \). Subtract 2 from both sides to transform it into \( a^2 - 6a - 2 = 0 \).

Step 2: Identify coefficients

Identify the coefficients from the equation \( a^2 - 6a - 2 = 0 \). Here, \( a = 1 \), \( b = -6 \), and \( c = -2 \).

Step 3: Apply the quadratic formula

The quadratic formula is \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Substitute the values of \( a \), \( b \), and \( c \) (which are 1, -6, and -2 respectively) into the formula:\[a = \frac{-(-6) \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot (-2)}}{2 \cdot 1}\]

Step 4: Simplify under the square root

Calculate the discriminant \( b^2 - 4ac \):\((-6)^2 = 36\) and \(4 \cdot 1 \cdot (-2) = -8\).This gives \(36 + 8 = 44\). So, \(\sqrt{44} = \sqrt{4 \times 11} = 2\sqrt{11}\).

Step 5: Compute the solutions

Plug the value of the square root back into the equation:\[a = \frac{6 \pm 2\sqrt{11}}{2}\].Simplify this to find the solutions:\[a = 3 \pm \sqrt{11}\].So, the solutions are \(a_1 = 3 + \sqrt{11}\) and \(a_2 = 3 - \sqrt{11}\).

Step 6: Verify using the sum and product of roots

For the quadratic equation \(ax^2 + bx + c = 0\):1. The sum of the roots \( (-b/a) \) is \(6/1 = 6\).2. The product of the roots \( c/a \) is \(-2/1 = -2\).Check: \( (3 + \sqrt{11}) + (3 - \sqrt{11}) = 6 \) and \( (3 + \sqrt{11})(3 - \sqrt{11}) = 9 - 11 = -2 \).Both the sum and product are correct.

Key Concepts

The Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It allows you to find the solutions, also known as the roots of the equation, by plugging the coefficients \(a\), \(b\), and \(c\) into this magical formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]This formula neatly handles all kinds of quadratic equations, including those with real or complex solutions. It's important first to rewrite the quadratic equation in standard form so that you can easily identify the coefficients. Once you substitute these values into the formula, you can find the roots. The plus-minus symbol \(\pm\) indicates that there might be two solutions: one involving addition and the other subtraction. This results from the square root operation.

Understanding the Discriminant

The discriminant is part of the quadratic formula and is denoted by \(b^2 - 4ac\). It gives crucial information about the nature of the roots of a quadratic equation.

• If the discriminant is positive, the equation has two distinct real roots.
• If it's zero, there is exactly one real root, also known as a repeated or double root.
• If the discriminant is negative, there are two complex roots.

In our original problem, we calculated the discriminant to be 44. Since this is positive, it confirms that we have two distinct real roots. Understanding the discriminant helps you predict the behavior of the solutions without necessarily solving the entire equation.

Sum and Product of Roots

The properties of the sum and product of roots are useful tools for verifying your solutions to a quadratic equation. Based on the standard form \(ax^2 + bx + c = 0\):

• The sum of the roots is equal to \(\frac{-b}{a}\).
• The product of the roots is equal to \(\frac{c}{a}\).

In the exercise, the sum and product were checked to be accurate with \(6\) and \(-2\), respectively. The sum is evaluated by adding the found roots: \((3+\sqrt{11}) + (3-\sqrt{11}) = 6\). The product is given by \((3+\sqrt{11})(3-\sqrt{11}) = 9 - 11 = -2\). These properties not only validate the solutions but also help understand the structural arrangement of roots.

Clarifying Polynomial Equations

Polynomial equations involve variables raised to natural number exponents. A quadratic equation, specifically, is a second-degree polynomial equation. It is characterized by the x-variable being squared (x^2). Quadratics can model a variety of real-world situations, from projectile motion to economics. Working with polynomial equations often involves:

• Rewriting the equation in standard form where needed.
• Identifying coefficients to use specific solving formulas, like the quadratic formula.
• Factoring when possible for simpler equations.

By mastering these techniques, you’re better equipped to approach more complex algebraic problems. Understanding the basic nature of quadratics as polynomials simplifies learning how to handle higher degree polynomial equations.