Problem 36 - Intermediate Algebra

Question

Simplify and reduce each expression. $$ 6 x^{2}-4 x=3 $$

Step-by-Step Solution

Answer

Solutions are \( x = \frac{2 + \sqrt{22}}{6} \) and \( x = \frac{2 - \sqrt{22}}{6} \).

Step 1: Move Terms to One Side

First, let's move all terms to one side of the equation to create a quadratic equation. Subtract 3 from both sides of the equation: \[ 6x^2 - 4x - 3 = 0 \]

Step 2: Identify Quadratic Equation Components

This equation is in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). Identify the coefficients: \( a = 6 \), \( b = -4 \), and \( c = -3 \).

Step 3: Use the Quadratic Formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Substitute \( a = 6 \), \( b = -4 \), and \( c = -3 \) into the formula:

Step 4: Compute the Discriminant

Calculate the discriminant \( b^2 - 4ac \):\[ (-4)^2 - 4(6)(-3) = 16 + 72 = 88 \].

Step 5: Solve for Roots

Substitute the discriminant back into the quadratic formula:\[ x = \frac{4 \pm \sqrt{88}}{12} \]Simplify the square root and the fraction:\[ x = \frac{4 \pm 2\sqrt{22}}{12} = \frac{2(2 \pm \sqrt{22})}{12} = \frac{2 \pm \sqrt{22}}{6} \].

Step 6: State the Simplified Roots

The solutions for \( x \) in the simplified form are:\[ x = \frac{2 + \sqrt{22}}{6} \quad \text{and} \quad x = \frac{2 - \sqrt{22}}{6} \].

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool used in algebra to find solutions for quadratic equations. A quadratic equation is generally expressed in the standard form as \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are known as coefficients, and \( a \) is not equal to zero. This formula allows us to solve for \( x \), the variable, even when the equation does not easily factor.The formula we use is:

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

The steps involve identifying the coefficients from your quadratic equation and substituting them into the formula. Here, \( \pm \) indicates there are typically two solutions, which are called the roots of the equation.Using the quadratic formula requires careful substitution and simplification to arrive at the correct roots, as any small mistake can lead to the wrong result.

Discriminant

The discriminant is a key part of determining the number and type of solutions in a quadratic equation. In the quadratic formula, the expression inside the square root \( b^2 - 4ac \) is known as the discriminant. It provides insight into the nature of the roots without needing to find them entirely.Let's break down what the discriminant reveals:

• If the discriminant is positive, there are two distinct real roots. This means the parabola representing the quadratic equation intersects the x-axis at two points.
• If the discriminant is zero, there is exactly one real root, or the roots are repeated. This occurs when the parabola just touches the x-axis.
• If the discriminant is negative, there are no real roots, but rather two complex roots. In this case, the parabola does not intersect the x-axis at all.

In the exercise, with a discriminant of 88, we knew to expect two real and distinct roots. Evaluating the discriminant helps decide if further simplification using the quadratic formula is needed.

Simplifying Expressions

Simplifying expressions, particularly those from quadratic equations, involves reducing the equation into its simplest form. This process includes algebraic manipulation to make the expression or the solution more understandable and concise.When you solve a quadratic equation using the quadratic formula, the result often needs simplification:

• Simplify any radicals (square roots) to their simplest form.
• Reduce fractions to make sure they are in their lowest terms.

For example, in the exercise, the square root of 88 was simplified to \( 2\sqrt{22} \), and the fraction \( \frac{4 \pm 2\sqrt{22}}{12} \) was reduced to \( \frac{2 \pm \sqrt{22}}{6} \). These simplifications make it easier to understand and use the solutions further.