Problem 4 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-14: Solve the equation by factoring. $$ 15 x^{2}-14=29 x $$

Step-by-Step Solution

Answer

The solutions are \( x = -\frac{2}{5} \) and \( x = \frac{7}{3} \).

Step 1: Write the equation in standard form

Start by rearranging the equation to get all the terms on one side of the equation. The given equation is \( 15x^2 - 14 = 29x \). Move the \( 29x \) to the left side by subtracting it from both sides: \( 15x^2 - 29x - 14 = 0 \). This is the standard quadratic form \( ax^2 + bx + c = 0 \).

Step 2: Identify the coefficients

Identify the coefficients in the equation \( 15x^2 - 29x - 14 = 0 \). Here, \( a = 15 \), \( b = -29 \), and \( c = -14 \).

Step 3: Factor the quadratic equation

To factor the quadratic equation, look for two numbers that multiply to \( a \times c = 15 \times -14 = -210 \) and add to \( b = -29 \). These numbers are \(-35\) and \(6\) because \(-35 \times 6 = -210\) and \(-35 + 6 = -29\). Rewrite the equation as \( 15x^2 - 35x + 6x - 14 = 0 \).

Step 4: Group and factor by grouping

Group the terms in pairs: \( (15x^2 - 35x) + (6x - 14) = 0 \). Factor out the greatest common factor from each pair: \( 5x(3x - 7) + 2(3x - 7) = 0 \). Notice that \(3x - 7\) is a common factor. Factor it out: \((5x + 2)(3x - 7) = 0\).

Step 5: Set each factor equal to zero

To solve for \( x \), set each factor equal to zero: \(5x + 2 = 0\) and \(3x - 7 = 0\).

Step 6: Solve each equation

Solve the equations: For \(5x + 2 = 0\), subtract 2 from both sides to get \(5x = -2\), then divide by 5 to get \(x = -\frac{2}{5}\). For \(3x - 7 = 0\), add 7 to both sides to get \(3x = 7\), then divide by 3 to get \(x = \frac{7}{3}\).

Key Concepts

Quadratic Equation

A quadratic equation is a type of polynomial equation of degree two. This means its highest exponent of the variable is 2. The general form of a quadratic equation is given by \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The quadratic equation is fundamental in algebra and appears in various mathematical problems.

• '\( a \)' is the coefficient of \( x^2 \), and it determines the width and direction of the parabola formed by the quadratic equation when graphed.
• '\( b \)' is the coefficient of \( x \), affecting the symmetry and position of the parabola on the x-axis.
• '\( c \)' is the constant term, which shifts the parabola up or down on the y-axis.

Quadratic equations are solved using various methods, such as factoring, completing the square, and the quadratic formula. Understanding these methods allows students to solve quadratic equations and apply them to real-world scenarios.

Factoring by Grouping

Factoring by grouping is a key method used to solve quadratic equations that cannot be factored easily with simple factor pairs. This technique involves rearranging and grouping terms in a polynomial to identify common factors for each pair, ultimately simplifying the expression into factors.
To factor by grouping, follow these steps:

• Rearrange the equation, if necessary, to make finding correct groupings easier.
• Split the middle term into two separate terms that facilitate grouping. This often involves finding two numbers that multiply to the product of the leading coefficient and the constant term, and add up to the middle coefficient.
• Group terms into pairs.
• Factor out the greatest common factor (GCF) from each pair of terms.
• Check for a common binomial factor and factor it out.

This technique simplifies a quadratic equation into a product of binomials, making it easier to solve by setting each factor equal to zero.

Polynomial Coefficients

Polynomial coefficients are the numbers in front of the variables within a polynomial expression. They play an essential role in determining the characteristics and solutions of the polynomial equation.
For a quadratic equation \( ax^2 + bx + c = 0 \), the coefficients are:

• \( a \): This is the quadratic coefficient. It's crucial as it affects the direction (upward or downward) and the width of the parabola when plotted. If \( a \) is positive, the parabola opens upwards, and if negative, it opens downwards.
• \( b \): The linear coefficient affects the symmetry and the horizontal alignment of the parabola's vertex.
• \( c \): The constant term influences the vertical position of the parabola since it represents the y-intercept where the graph crosses the y-axis.

Correctly identifying these coefficients makes it easier to choose and execute appropriate strategies for solving the quadratic equation. Understanding their impact on the shape and position of the parabola aids in a deeper comprehension of quadratic functions.