Question

The equation formed by decreasing each root of \(a x^{2}+b x+c=0\) by 1 is \(2 x^{2}+8 x+2=0\). Then, (a) \(a=-b\) (b) \(\mathrm{b}=-\mathrm{c}\) (c) \(\mathrm{c}=-\mathrm{a}\) (d) \(b=a+c\)

Short answer

Based on the analysis and solution, the correct answer is: (d) \(b = a + c\)

Step-by-step solution

Calculate the sum and the product of the roots of the original equation

Let the roots of the original equation be \(r_1\) and \(r_2\). According to Vieta's formulas, the sum of the roots is given by \(-(\frac{b}{a})\) and the product by \(\frac{c}{a}\). So, we have: \(r_1 + r_2 = -\frac{b}{a}\) \(r_1 \cdot r_2 = \frac{c}{a}\)

Calculate the sum and the product of the roots of the modified equation

Now, let's call the roots of the modified equation \(r'_1\) and \(r'_2\). Since the modified equation is obtained by decreasing each root of the original equation by 1, we have: \(r'_1 = r_1 - 1\) \(r'_2 = r_2 - 1\) Now, find the sum and product of \(r'_1\) and \(r'_2\) using Vieta's formulas for the modified equation \(2x^2+8x+2=0\). \(r'_1 + r'_2 = -\frac{8}{2} = -4\) \(r'_1 \cdot r'_2 = \frac{2}{2} = 1\)

Analyze the relationships using the sums and products of the roots

Now, we want to find a relationship between \(a\), \(b\), and \(c\). We start by analyzing the relationships using the sums and products of the roots. We have 4 equations from steps 1 and 2: \(r_1 + r_2 = -\frac{b}{a}\) \(r_1 \cdot r_2 = \frac{c}{a}\) \(r'_1 + r'_2 = -4\) \(r'_1 \cdot r'_2 = 1\) From the third equation, we get: \(r'_1 + r'_2 = (r_1 - 1) + (r_2 - 1) = r_1 + r_2 - 2 = -4\) Plugging in the sum of the roots from the first equation, we have: \(-\frac{b}{a} - 2 = -4\) Now, solve for \(b\): \(b = 2a\) Next, we look at the relationship between the products: \(r'_1 \cdot r'_2 = (r_1 - 1)(r_2 - 1) = 1\) Expanding and using the fact that \(r_1 \cdot r_2 = \frac{c}{a}\), we get: \(r_1 \cdot r_2 - r_1 - r_2 + 1 = \frac{c}{a} - \frac{b}{a} + 1 = 1\) Simplify: \(c - b + a = a^2\) Now, use the relationship \(b = 2a\) obtained earlier: \(c - 2a + a = a^2\) \(c - a = a^2\)

Determine the correct answer

To determine the correct answer from the choices provided, compare each statement with relations we have found. (a) \(a = -b\) - This is incorrect because we found that \(b = 2a\). (b) \(b = -c\) - This is incorrect because we found that \(c - a = a^2\). (c) \(c = -a\) - This is incorrect because we found that \(c - a = a^2\). (d) \(b = a + c\) - This is correct because if we rewrite our relation \(c - a = a^2\) as \(c = a(a + 1)\), we can replace \(b\) in the choice with our \(b = 2a\) relation and get \(2a = a + a(a + 1)\). So, the answer is (d).

Key concepts

Vieta's Formulas

Vieta's formulas offer a simple way to understand the relationships between the coefficients of a polynomial and its roots. For quadratic equations of the form \(ax^2 + bx + c = 0\), Vieta's formulas state:

• The sum of the roots \((r_1 + r_2)\) is \(-\frac{b}{a}\).
• The product of the roots \((r_1 \cdot r_2)\) is \(\frac{c}{a}\).

These formulas are very helpful because they allow us to quickly express the roots in terms of the coefficients without actually finding their specific values. In the exercise, we used Vieta's formulas to derive relationships involving \(a\), \(b\), and \(c\) by transforming the original roots.

Roots Transformation

Roots transformation involves altering the roots of an equation and then studying the effects on the equation's coefficients. In the given problem, we decreased each root by 1:

• If the original roots were \(r_1\) and \(r_2\), then the transformed roots are \(r_1' = r_1 - 1\) and \(r_2' = r_2 - 1\).

This procedure leads to a new quadratic equation, \(2x^2 + 8x + 2 = 0\). Using Vieta's formulas again, we found the sum and product of the transformed roots, \(r_1' + r_2'\) and \(r_1' \cdot r_2'\). The transformation allowed us to derive relationships between the coefficients of the original and modified equations.

Quadratic Equation Relationships

Understanding the relationships between the coefficients of quadratic equations is crucial. By examining the sum and product of the roots, we can solve for missing variables. In the problem:

• We found \(r_1 + r_2 = -\frac{b}{a}\) and \(r_1 \cdot r_2 = \frac{c}{a}\).
• After transforming the roots, \(r_1' + r_2' = -4\) and \(r_1' \cdot r_2' = 1\).
• These led us to \(b = 2a\), linking \(b\) and \(a\), and \(c - a = a^2\), linking \(c\) and \(a\).

By solving these equations, we verified the correct relationship \(b = a + c\), confirming option (d) as the right answer. This process highlights the power of using root properties to uncover the dynamics between coefficients.