Problem 33 - Algebra A Combined Function

Question

Solve. $$ x(3 x-1)=14 $$

Step-by-Step Solution

Answer

The solutions to the equation are \( x = 2.333 \) and \( x = -2 \).

Step 1: Distribute the x

Start by distributing the variable \( x \) to the terms within the parentheses. Rewrite the equation as: \[ 3x^2 - x = 14 \]

Step 2: Move all terms to one side

To form a standard quadratic equation, subtract 14 from both sides. This gives us: \[ 3x^2 - x - 14 = 0 \]

Step 3: Use the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). For our equation, \( a = 3 \), \( b = -1 \), and \( c = -14 \). Substitute these into the formula: \[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 3 \cdot (-14)}}{2 \cdot 3} \]

Step 4: Simplify under the square root

Calculate the discriminant (\( b^2 - 4ac \)): \[ (-1)^2 - 4 \cdot 3 \cdot (-14) = 1 + 168 = 169 \] Since \( 169 \) is a perfect square, \( \sqrt{169} = 13 \).

Step 5: Complete the calculations

Substitute \( \,\pm 13 \) back into the quadratic formula to find x: \[ x = \frac{1 \pm 13}{6} \] Calculate the two possible values of \( x \): \[ x_1 = \frac{1 + 13}{6} = 2.333 \] \[ x_2 = \frac{1 - 13}{6} = -2 \]

Step 6: Verify the solutions

Verify each solution by substituting back into the original equation: For \( x = 2.333 \), \( 2.333(3 \times 2.333 - 1) \approx 14 \) (confirm by actual multiplication). For \( x = -2 \), \( -2(3 \times -2 - 1) = 14 \). Both solutions satisfy the equation.

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which are equations of the form \( ax^2 + bx + c = 0 \). The formula is expressed as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula allows you to find the values of \( x \) that make the equation true. It works for any quadratic equation, helping you find the roots efficiently.

• Steps to Use: Identify the coefficients \( a \), \( b \), and \( c \) from your equation. These are usually the numbers next to \( x^2 \), \( x \), and the constant term, respectively.
• Substitute The Values: Once you have \( a \), \( b \), and \( c \), substitute them into the formula to calculate the roots.
• Calculate Two Solutions: The \( \pm \) sign means you'll perform the calculation twice - once using addition and once using subtraction. This results in two possible values for \( x \).

Keep in mind that the quadratic formula can always be applied, even when other simpler methods might not work due to complex numbers or non-factorable polynomials.

Discriminant

The discriminant is a part of the quadratic formula, represented by \( b^2 - 4ac \). It determines the nature and number of solutions of a quadratic equation.

• Discriminant Value: If the discriminant is positive, you get two real and distinct solutions. If it is zero, there is exactly one real solution (or a repeated root). A negative discriminant indicates complex or imaginary solutions.
• Example in Context: In the equation from the original exercise, the discriminant is calculated as \((-1)^2 - 4 \times 3 \times (-14) = 169\). This positive value tells us that there are two distinct real solutions.

The discriminant provides quick insights into the possible nature of the solutions, allowing you to anticipate the results before completing all your calculations.

Perfect Square

A perfect square is an integer that is the square of another integer. For example, 1, 4, 9, 16, etc., are perfect squares because they result from squaring 1, 2, 3, 4, respectively. Recognizing perfect squares is helpful in the quadratic formula.

• Connection to Quadratics: When the discriminant (\( b^2 - 4ac \)) is a perfect square, the square root of the discriminant is a simple integer. This simplifies your calculations significantly.
• Application in Exercise: From our exercise, the discriminant is 169, which is a perfect square since \( \, \sqrt{169} = 13 \).

Identifying perfect squares helps in quickly solving quadratic equations without dealing with complex square root calculations.

Standard Quadratic Equation

A standard quadratic equation takes the form \( ax^2 + bx + c = 0 \). It's essential for ensuring consistency with methods and formulas like factoring, completing the square, and using the quadratic formula.

• Reformulating Questions: Convert any quadratic problem into this standard form by moving all terms to one side and arranging them in descending order of power.
• Identifying Coefficients: With the equation in this form, you can easily spot and label coefficients \( a \), \( b \), and \( c \), which are crucial for using the quadratic formula and determining the discriminant.

Establishing the equation in standard form is a vital first step in applying any algebraic technique to solve it efficiently.