Problem 27 - Intermediate Algebra

Question

Simplify and reduce each expression. $$ 3 a^{2}-8 a+2=0 $$

Step-by-Step Solution

Answer

The simplified solutions for \( a \) are \( \frac{4}{3} + \frac{\sqrt{10}}{3} \) and \( \frac{4}{3} - \frac{\sqrt{10}}{3} \).

Step 1: Identify the Type of Equation

First, recognize that the given equation \( 3a^2 - 8a + 2 = 0 \) is a quadratic equation, as it is in the form \( ax^2 + bx + c = 0 \) where \( a = 3 \), \( b = -8 \), and \( c = 2 \).

Step 2: Apply the Quadratic Formula

The quadratic formula is \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). We will use this formula to find the roots of the equation. Substitute \( a = 3 \), \( b = -8 \), and \( c = 2 \) into the formula.

Step 3: Calculate the Discriminant

The discriminant is given by \( b^2 - 4ac \). Substitute \( b = -8 \), \( a = 3 \), and \( c = 2 \) into the equation: \((-8)^2 - 4 \times 3 \times 2 = 64 - 24 = 40 \).

Step 4: Substitute Back into the Quadratic Formula

Now substitute the values into the quadratic formula: \( a = \frac{-(-8) \pm \sqrt{40}}{2 \times 3} \), which simplifies to \( a = \frac{8 \pm \sqrt{40}}{6} \).

Step 5: Simplify the Expression

Simplify \( \sqrt{40} \) which is \( \sqrt{4 \times 10} = 2\sqrt{10} \). This results in \( a = \frac{8 \pm 2\sqrt{10}}{6} \).

Step 6: Final Simplification

Lastly, break apart the fractions: \( a = \frac{8}{6} \pm \frac{2\sqrt{10}}{6} \). Simplifying, this becomes \( a = \frac{4}{3} \pm \frac{\sqrt{10}}{3} \).

Key Concepts

Quadratic Formula

Quadratic equations appear in the form of \( ax^2 + bx + c = 0 \). They are specific polynomial equations of the second degree. This signifies that the highest power of the variable—in this case, \( a \)—is squared. Solving these equations can sometimes be tricky. This is where the quadratic formula steps in to help. The quadratic formula is an essential tool used to find the roots of any quadratic equation. The formula itself is quite friendly and is expressed as:

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

This formula lets you solve for \( x \) by simply plugging in the values of \( a \), \( b \), and \( c \) from your quadratic equation. Here, "\( b^2 - 4ac \)" is known as the discriminant, which we will discuss in the next section.Once you have these values substituted into the formula, solving for \( x \) becomes a straightforward process. Using the quadratic formula can save time and avoid errors that might arise from other methods like factoring or completing the square. It's particularly useful when the equation does not factorize easily.

Discriminant Calculation

The discriminant is a key element when using the quadratic formula. It helps determine how many and what type of solutions the quadratic equation will have. The discriminant is represented by the part of the quadratic formula under the square root: \( b^2 - 4ac \).Here's why it's important:

• If \( b^2 - 4ac > 0 \), there are two distinct real roots.
• If \( b^2 - 4ac = 0 \), there is exactly one real root (or a repeated root).
• If \( b^2 - 4ac < 0 \), there are no real roots, but two complex roots.

For the given equation \( 3a^2 - 8a + 2 = 0 \), we calculated the discriminant as 40. Since this value is positive, it indicates that there are two distinct real solutions to our quadratic equation. This tells us that the solutions will be real and different, which guides us in simplifying the expression.

Simplifying Radicals

Once you've substituted into the quadratic formula, your next step involves simplifying radicals, particularly in the term \( \sqrt{b^2 - 4ac} \). In any case where the discriminant is not a perfect square, like \( 40 \) in this instance, you'll need to simplify the square root for further reduction.To simplify \( \sqrt{40} \), you start by factoring it into a product of integers: \( \sqrt{4 \times 10} \). Breaking down further, you find \( \sqrt{4} \) is \( 2 \), thus simplifying the entire expression to \( 2\sqrt{10} \).The simplification does not stop here. You'll also want to separate your fractions, such as bringing \( a = \frac{8 \pm 2\sqrt{10}}{6} \) into individual parts. Divide each component separately:

• For the constant term, \( \frac{8}{6} \) simplifies to \( \frac{4}{3} \).
• For the term with the radical, \( \frac{2\sqrt{10}}{6} \) simplifies to \( \frac{\sqrt{10}}{3} \).

By breaking down and simplifying these expressions, you achieve a clean and reduced form: \( a = \frac{4}{3} \pm \frac{\sqrt{10}}{3} \). This makes it easier to understand the roots of the equation.