Problem 6 - Algebra and Trigonometry

Question

Solve the equation by factoring. \(4 x^{2}-4 x-15=0\)

Step-by-Step Solution

Answer

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

Step 1: Identify the equation to factor

The equation we have is \(4x^2 - 4x - 15 = 0\). This is a quadratic equation in the form \(ax^2 + bx + c = 0\) with \(a = 4\), \(b = -4\), and \(c = -15\).

Step 2: Multiplying to find factor pairs

We need to factor the quadratic expression. The first step is to multiply the coefficient of \(x^2\), which is 4, by the constant term, which is -15. That gives us \(4 imes (-15) = -60\). We are looking for two numbers that multiply to -60 and add to the coefficient of \(x\), which is -4.

Step 3: Finding the pair of numbers

The numbers that multiply to -60 and add to -4 are 6 and -10.

Step 4: Rewrite the middle term

Use the numbers found in Step 3 to rewrite the equation: \(4x^2 - 10x + 6x - 15 = 0\).

Step 5: Factor by grouping

Next, group the terms and factor each group separately: - Group 1: \(4x^2 - 10x\) can be factored as \(2x(2x - 5)\). - Group 2: \(6x - 15\) can be factored as \(3(2x - 5)\). So we have: \(2x(2x - 5) + 3(2x - 5) = 0\).

Step 6: Factor out the common binomial

Now, factor out the common factor, \((2x - 5)\): \((2x - 5)(2x + 3) = 0\).

Step 7: Solve for x

Set each factor equal to zero and solve for \(x\):- \(2x - 5 = 0\) gives \(x = \frac{5}{2}\). - \(2x + 3 = 0\) gives \(x = -\frac{3}{2}\).

Key Concepts

Solving Quadratic Equations

Solving quadratic equations is a fundamental skill in algebra. These equations take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) represents an unknown value. Quadratic equations are typically solved using methods such as:

• Factoring
• Using the quadratic formula
• Completing the square

For this exercise, the focus is on factoring, which involves rewriting the quadratic equation as a product of simpler expressions. The solutions, or "roots," are the values of \( x \) that satisfy the equation. Identifying these roots helps in understanding relationships between variables in real-world scenarios, like physics, engineering, or finance.

Factoring by Grouping

Factoring by grouping is a method used to factor polynomials by splitting a middle term into two terms, based on specific criteria. This technique is especially useful in quadratic equations like \( 4x^2 - 4x - 15 = 0 \). In this method, you rewrite the quadratic equation in a form that allows for rearranging terms in pairs

• Find two numbers that multiply to \( a \cdot c \) and add to \( b \)
• Rewrite the quadratic with these new terms inserted, creating four terms
• Group terms two-by-two and factor the greatest common factors from each
• Factor out the common binomial factor from the two grouped expressions

This approach effectively breaks down the polynomial into factored components, making solving for \( x \) straightforward.

Quadratic Formula

The quadratic formula is a powerful tool for solving any quadratic equation. It can always be used, even when other methods like factoring or completing the square are not feasible. The formula is given as:\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]This formula calculates the roots by plugging in the values of \( a \), \( b \), and \( c \) from the standard form of the quadratic equation \( ax^2 + bx + c = 0 \). Using the quadratic formula involves four essential steps:

• Identifying coefficients \( a \), \( b \), and \( c \)
• Calculating the discriminant \( b^2 - 4ac \)
• Applying the values in the quadratic formula
• Simplifying to find the roots

Its flexibility and providing exact solutions make the quadratic formula invaluable for students and professionals alike.

Roots of Equations

The roots of a quadratic equation are the values of \( x \) that make the equation true, meaning they satisfy \( ax^2 + bx + c = 0 \). These roots can be real or complex numbers, depending on the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there is exactly one real root (a repeated root).
• If the discriminant is negative, the equation has two complex roots.

Finding the roots is an essential part of solving quadratic equations, providing insights into the graph of the equation, its vertex and intercepts. For example, the quadratic \( 4x^2 - 4x - 15 = 0 \) has roots \( x = \frac{5}{2} \) and \( x = -\frac{3}{2} \), which are the points where the parabola crosses the x-axis.