Problem 2 - Algebra 2 and Trigonometry

Question

Adrien said that if the roots of a quadratic equation are \(\frac{1}{2}\) and \(\frac{3}{4}\) , the equation is \(4 x^{2}-5 x+\frac{3}{2}=0 .\) Olivia said that the equation is \(8 x^{2}-10 x+3=0 .\) Who is correct? Justify your answer.

Step-by-Step Solution

Answer

Olivia's equation is correct because both the sum and product of the given roots match her equation's coefficients.

Step 1: Understanding the problem

We need to determine which quadratic equation, proposed by Adrien or Olivia, is correct given the roots \( \frac{1}{2} \) and \( \frac{3}{4} \). In a quadratic equation, the sum and product of the roots are related to the coefficients. Let's check both equations.

Step 2: Check the Sum and Product of Roots

For a quadratic equation \( ax^2 + bx + c = 0 \) with roots \( p \) and \( q \), the sum of the roots \( p + q = -\frac{b}{a} \), and the product \( pq = \frac{c}{a} \). For both proposed equations, let's calculate and compare these values to the given roots.

Step 3: Calculating for Adrien's Equation

Adrien's equation is \( 4x^2 - 5x + \frac{3}{2} = 0 \). Here, \( a = 4, b = -5, c = \frac{3}{2} \). The sum of the roots is \( \frac{1}{2} + \frac{3}{4} = \frac{5}{4} \). We compare this to \( -\frac{b}{a} = \frac{5}{4} \). The product is \( \frac{3}{2} \times 1 = \frac{3}{2} \), which matches \( \frac{c}{a} = \frac{3}{2} / 4 = \frac{3}{8} \), so this does not match.

Step 4: Calculating for Olivia's Equation

Olivia's equation is \( 8x^2 - 10x + 3 = 0 \). Here, \( a = 8, b = -10, c = 3 \). The sum of the roots \( \frac{1}{2} + \frac{3}{4} = \frac{5}{4} \) matches \( -\frac{b}{a} = \frac{10}{8} = \frac{5}{4} \). The product of the roots \( \frac{1}{2} \times \frac{3}{4} = \frac{3}{8} \) matches \( \frac{c}{a} = \frac{3}{8} \). Olivia is, therefore, correct.

Key Concepts

Roots of a Quadratic

In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the values of \( x \) that satisfy the equation are called the roots. These roots can be real or complex numbers. The quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) can be used to find the roots of any quadratic equation.
The nature of the roots depends on the discriminant \( b^2 - 4ac \):

• If the discriminant is positive, there are two distinct real roots.
• If the discriminant is zero, there are two identical real roots.
• If the discriminant is negative, the roots are complex and not real numbers.

Understanding how to manipulate and solve for these roots is crucial in determining the form of the quadratic equation. Various methods, such as factoring, completing the square, or the quadratic formula, help identify these roots.

Sum and Product of Roots

The sum and product of the roots of a quadratic equation relate to its coefficients.
For an equation \( ax^2 + bx + c = 0 \) with roots \( p \) and \( q \), the sum of the roots \( p + q \) can be found using the formula \(-\frac{b}{a}\), while the product \( pq \) is equal to \( \frac{c}{a} \).

• The sum formula \( p + q = -\frac{b}{a} \) indicates that the sum of the roots correlates with the linear coefficient and is inverted in sign.
• The product formula \( pq = \frac{c}{a} \) contrasts the relation between the roots and the constant term, modified by the leading coefficient.

Knowing these relations is valuable in both finding the roots and constructing an equation when the roots are given. This was essential in verifying which quadratic provided the correct roots in our original problem.

Quadratic Equation Coefficients

The coefficients \( a, b, \) and \( c \) in the equation \( ax^2 + bx + c = 0 \) play pivotal roles in shaping its graph and nature.

• \(a\): The leading coefficient, dictates the parabola's direction of opening. A positive \(a\) opens upward, while a negative \(a\) opens downward.
• \(b\): This coefficient, in conjunction with \(a\), affects the position and symmetry of the parabola.
• \(c\): The constant term marks the y-intercept of the quadratic graph.

To determine the equation when roots are provided, one must rearrange the standard form to reflect these roots and coefficients. Moreover, understanding the relationship between these coefficients and the roots helps to verify or construct quadratic equations accurately, as demonstrated in selecting Olivia's equation over Adrien's.