Problem 11 - Algebra 2 and Trigonometry

Question

Without solving each equation, find the sum and product of the roots. \(2 x^{2}-8=5 x\)

Step-by-Step Solution

Answer

Sum of roots: \( \frac{5}{2} \), Product of roots: \(-4\).

Step 1: Write the Quadratic in Standard Form

Start by rearranging the given equation into standard quadratic form, which is \( ax^2 + bx + c = 0 \). The original equation is \( 2x^2 - 8 = 5x \). Rearrange this to get \( 2x^2 - 5x - 8 = 0 \).

Step 2: Identify Coefficients

Identify the coefficients from the standard form \( ax^2 + bx + c = 0 \). Here we have \( a = 2 \), \( b = -5 \), and \( c = -8 \).

Step 3: Sum of the Roots

According to Vieta's formulas, the sum of the roots \( r_1 + r_2 \) of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( -\frac{b}{a} \). Substitute the values of \( b \) and \( a \) to find the sum: \( r_1 + r_2 = -\frac{-5}{2} = \frac{5}{2} \).

Step 4: Product of the Roots

Also from Vieta's formulas, the product of the roots \( r_1 \times r_2 \) is given by \( \frac{c}{a} \). Substitute the values of \( c \) and \( a \) to find the product: \( r_1 \times r_2 = \frac{-8}{2} = -4 \).

Key Concepts

Quadratic Equations

Quadratic equations are a type of polynomial equation that are characterized by the presence of the term with the variable raised to the second power. In standard form, a quadratic equation looks like this: \( ax^2 + bx + c = 0 \). Here, \( a \), \( b \), and \( c \) are coefficients, and \( a \) cannot be zero, as that would make the equation linear instead of quadratic. The solution to a quadratic equation is typically found by solving for the variable, \( x \), which can have up to two real roots. These roots represent the values of \( x \) that make the equation true.

Sum of Roots

The sum of the roots of a quadratic equation is a unique relationship defined by Vieta's formulas. Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. In the context of a quadratic equation \( ax^2 + bx + c = 0 \), the sum of the roots, \( r_1 + r_2 \), can be found using the formula:

• \( r_1 + r_2 = -\frac{b}{a} \)

This formula implies that the sum of the roots of the quadratic equation is equal to the negative of the coefficient of the linear term divided by the coefficient of the quadratic term. This relation gives us a direct way to find the sum of the roots without actually solving the equation.

Product of Roots

Just like the sum of the roots, the product of the roots of a quadratic equation is also defined by Vieta's formulas. For the standard quadratic equation \( ax^2 + bx + c = 0 \), the product of the roots \( r_1 \cdot r_2 \) is given by:

• \( r_1 \cdot r_2 = \frac{c}{a} \)

This formula states that the product of the roots is equal to the constant term, \( c \), divided by the coefficient of \( x^2 \), \( a \). This relationship provides a straightforward way to calculate the product of the roots using the equation's coefficients without solving the equation for its roots.

Coefficients in Quadratic Equations

In the quadratic equation \( ax^2 + bx + c = 0 \), the coefficients \( a \), \( b \), and \( c \) play crucial roles. Each coefficient represents a specific component of the equation:

• \( a \) is the coefficient of the \( x^2 \) term; it determines the parabola's direction (upward if \( a > 0 \), downward if \( a < 0 \)).
• \( b \) is the coefficient of the \( x \) term; it affects the parabola's symmetry and its vertex's horizontal position.
• \( c \) is the constant term; it influences where the parabola intersects the y-axis.

Understanding these coefficients allows us to use Vieta's formulas to find important properties of the roots, such as their sum and product, without needing to fully solve the equation.