Problem 9 - Algebra 2 and Trigonometry

Question

Without solving each equation, find the sum and product of the roots. \(8 x+12=x^{2}\)

Step-by-Step Solution

Answer

The sum of the roots is 8, and the product is -12.

Step 1: Rearrange the Equation

First, rearrange the given equation into standard quadratic form. The given equation is \( 8x + 12 = x^2 \). Rearrange it to form \( x^2 - 8x - 12 = 0 \).

Step 2: Identify Coefficients

In the quadratic equation \( ax^2 + bx + c = 0 \), identify coefficients \( a \), \( b \), and \( c \). Here, \( a = 1 \), \( b = -8 \), and \( c = -12 \).

Step 3: Calculate Sum of the Roots

The sum of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \(-\frac{b}{a}\). Therefore, the sum is \(-\frac{-8}{1} = 8\).

Step 4: Calculate Product of the Roots

The product of the roots of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \(\frac{c}{a}\). Therefore, the product is \(\frac{-12}{1} = -12\).

Key Concepts

Sum of Roots

In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the sum of the roots can be found without actually solving the equation. This is due to a property related to the coefficients of the equation. The formula for the sum of the roots is \(-\frac{b}{a}\).

Let's break this down step by step, using the exercise you provided. When you rearrange the equation \(8x + 12 = x^2\) to its standard form, it becomes \(x^2 - 8x - 12 = 0\). From here, you can see:

• \(a = 1\)
• \(b = -8\)
• \(c = -12\)

The sum of the roots is given by \(-\frac{b}{a} = -\frac{-8}{1} = 8\).

This simply means if you add both roots of this equation together, you'll get 8. This property is derived from Vieta's formulas, which relate the sum and product of the roots to the coefficients of the polynomial.

Product of Roots

Similar to finding the sum of the roots, you can also find the product of the roots without fully solving the quadratic equation. The formula for the product of the roots is \(\frac{c}{a}\).

Applying this to our example, we know from the standard form \(x^2 - 8x - 12 = 0\), the coefficients are:

• \(a = 1\)
• \(b = -8\)
• \(c = -12\)

The product of the roots is calculated by \(\frac{c}{a} = \frac{-12}{1} = -12\).

Thus, this tells us when you multiply the two roots, their product equals -12. This method provides significant insights into the nature of the roots without needing to solve for them directly and is again a part of Vieta’s formulas.

Standard Form of Quadratic Equation

In order to work with quadratic equations effectively, it helps to understand the standard form, which is \( ax^2 + bx + c = 0 \). This form helps simplify many processes, such as finding the sum and product of roots right away.

Let's go back to the initial equation \(8x + 12 = x^2\). To bring it to the standard form, we rearrange it to get \(x^2 - 8x - 12 = 0\).

This is important because:

• \(a\), \(b\), and \(c\) are clearly identified,
• Both the sum and product of the roots can easily be calculated once in this standard form.

With this approach, rather than focusing on solving the equation for its roots initially, converting to standard form provides the foundation for gleaning important information quickly. Having a firm grasp of the standard form makes learning and solving quadratic equations much more approachable.