Question

The condition for the sum and the product of the roots of the quadratic equation \(a x^{2}-b x+c=0\) to be equal, is (1) \(\mathrm{b}+\mathrm{c}=0\). (2) \(\mathrm{b}-\mathrm{c}=0\). (3) \(a+c=0\). (4) \(a+b+c=0\).

Short answer

Answer: The condition for the sum and the product of the roots of the quadratic equation \(ax^2 - bx + c = 0\) to be equal is \(b+c=0\).

Step-by-step solution

Write the Sum and Product of Roots Formulas

The sum and product of the roots of the quadratic equation \(ax^2 - bx + c = 0\) are given by the following formulas: Sum of roots: \(p + q = \frac{-b}{a}\) Product of roots: \(pq = \frac{c}{a}\)

Write the Condition for Equal Sum and Product of Roots

The condition that the sum and the product of the roots are equal, is: \(p + q = pq\) Using the formulas for the sum and product of roots from Step 1, we can write: \(\frac{-b}{a} = \frac{c}{a}\) We must find the expression that matches one of the four given options.

Simplify the Equation

To simplify the equation obtained in Step 2, we can start by multiplying both sides by \(a\), which results in: \(-b = c\)

Match the Condition with the Given Options

Looking at the given options, we see that the condition we derived in Step 3, \(-b = c\), matches option (1) \(\mathrm{b}+\mathrm{c}=0\). Therefore, the correct answer is option (1).

Key concepts

Sum of Roots

In any quadratic equation of the form \(ax^2 - bx + c = 0\), the sum of its roots can be easily calculated using a handy formula. Regardless of what the equation looks like in its entirety, the sum of its roots will be given by \(p + q = \frac{-b}{a}\). This formula is derived directly from the properties of polynomials, where \(p\) and \(q\) are the roots or solutions of the equation.
The importance of this formula lies in its simplicity; it gives a direct way to find the sum of the roots just by looking at the coefficients of the quadratic equation. Always remember that the coefficient \(b\) relates to the linear term, and that is part of the key in effectively using the formula provided.
Overall, understanding the sum of roots can significantly aid in solving various mathematical problems where quadratic equations are involved.

Product of Roots

Beyond just finding the sum, the product of roots of a quadratic equation is just as simple to determine. Using the equation \(ax^2 - bx + c = 0\) once more, the product of its roots can be determined by the equation \(pq = \frac{c}{a}\).
This equation tells us something fundamental about the relationship between the coefficients and the roots of the quadratic equation. In the case of our quadratic expression, \(c\) is a constant term, and knowing how it affects the roots through multiplication is crucial.
When trying to determine the product of the roots, this formula provides a shortcut and eliminates the need to handle the roots separately. Applying these formulas together can unveil a lot about the nature of equations and their solutions.

Roots of Quadratic Equations

Roots of a quadratic equation are the solutions that satisfy it. In mathematical terms, they are the values of \(x\) which make the equation true. For any quadratic equation \(ax^2 - bx + c = 0\), these roots can be identified as \(p\) and \(q\).
By the various algebraic strategies available, such as factoring, completing the square, or using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), these roots can be found. Each method has its utility, but the quadratic formula is universal, providing solutions that encompass both real and imaginary roots based on the discriminant \(b^2-4ac\).
Understanding these roots can offer insight into the behavior of quadratic functions, making them easier to graph and comprehend in various mathematical and practical contexts.

Equal Conditions for Roots

At times, the sum and the product of the roots of quadratic equations might be equal. This scenario provides a unique condition wherein \(p + q = pq\).
Using the sum and product of roots formulas, this condition can be rewritten as follows:

• \(\frac{-b}{a} = \frac{c}{a}\)

This implies that \(-b = c\) provides a direct condition that can help solve for specific relationships between the coefficients of the quadratic equation. When \(-b = c\), simplifying further gives \(b + c = 0\), which aligns perfectly with one of the solution conditions offered for the problem.
This insight not only simplifies problems but also enriches the understanding of how the relationships between coefficients affect the behavior of quadratic equations.