Problem 67 - Algebra 2

Question

Write a quadratic equation with the given roots. Write the equation in the form \(a x^{2}+b x+c=0,\) where \(a, b,\) and \(c\) are integers. \(-2,-7\)

Step-by-Step Solution

Answer

The quadratic equation is \(x^2 + 9x + 14 = 0\).

Step 1: Use Vieta's Formulas

For a quadratic equation \( ax^2 + bx + c = 0 \) with roots \( r_1 \) and \( r_2 \), Vieta's formulas tell us that the sum of the roots \( r_1 + r_2 = -\frac{b}{a} \) and the product of the roots \( r_1 \times r_2 = \frac{c}{a} \). Here, the given roots are \(-2\) and \(-7\). Using Vieta's, the sum of the roots is \(-2 + (-7) = -9\), and the product of the roots is \(-2 \times -7 = 14\).

Step 2: Formulate the Equation

Given that the roots are \(-2\) and \(-7\), we can substitute them directly into the factored form of a quadratic equation: \((x - r_1)(x - r_2) = 0\). This gives \((x + 2)(x + 7) = 0\).

Step 3: Expand the Equation

To find the quadratic equation in standard form, expand \((x + 2)(x + 7)\). This becomes \(x^2 + 7x + 2x + 14\).

Step 4: Combine Like Terms

Combine the like terms from the expansion. Thus, the equation becomes \(x^2 + 9x + 14 = 0\). This is in the form \(a x^2 + b x + c = 0\), where \(a = 1\), \(b = 9\), and \(c = 14\).

Key Concepts

Vieta's Formulas

Vieta's formulas are magical little tools that help you connect the dots between the coefficients of a polynomial and its roots. They provide a nifty shortcut when you're dealing with quadratic equations, or any polynomial, really. Simply put, these formulas help you understand:

• The sum of the roots is equal to the negation of the coefficient of the linear term divided by the coefficient of the quadratic term.
• The product of the roots is equal to the constant term divided by the coefficient of the quadratic term.

To illustrate, if you have a quadratic equation in the form \(ax^2 + bx + c = 0\), and the roots are \(r_1\) and \(r_2\), Vieta's formulas state:

• \(r_1 + r_2 = -\frac{b}{a}\)
• \(r_1 \times r_2 = \frac{c}{a}\)

In the exercise example, the roots \((-2)\) and \((-7)\) give us a sum of \(-9\) and a product of \(14\). These align perfectly with the coefficients derived from the expanded equation \(x^2 + 9x + 14 = 0\), confirming the power and accuracy of Vieta's formulas.

Roots of a Polynomial

The roots of a polynomial are simply the values that make the polynomial equal to zero. They are also known as the solutions to the polynomial equation. For a quadratic equation like \(ax^2 + bx + c = 0\), there can be two roots, which may be real or complex numbers.
In our exercise, the roots \(-2\) and \(-7\) mean that these are the values of \(x\) that satisfy the equation when substituted back into it. To find these numbers, you can work backwards from the factored form.

• Start with \((x - r_1)(x - r_2) = 0\), where \(r_1\) and \(r_2\) are roots.
• Substitute the given roots \(-2\) and \(-7\) to form \((x + 2)(x + 7) = 0\).

Upon expanding, you get the quadratic equation in the standard form: \(x^2 + 9x + 14 = 0\). This form tells you everything about the relationship between the polynomial's coefficients and its roots.

Standard Form of Quadratic Equation

When we talk about the standard form of a quadratic equation, we mean writing the equation as \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients, and \(a\) isn't zero. This form is the go-to way of presenting quadratic equations because it clearly shows all the necessary components.
In the given exercise, starting from the roots \(-2\) and \(-7\), we put them into the factored form and expanded to get \(x^2 + 9x + 14 = 0\). This is a textbook example of the standard form.

• \(a = 1\)
• \(b = 9\)
• \(c = 14\)

Each coefficient plays a role:

• \(a\) determines the direction and width of the parabola.
• \(b\) affects the placement and symmetry.
• \(c\) is the y-intercept, where the parabola crosses the y-axis.

By converting any quadratic expression into this form, you can easily apply Vieta's formulas and understand the polynomial's graph and intersection points with axes.