Question

The coefficient of \(\mathrm{x}\) in the quadratic equation \(\mathrm{x}^{2}+\mathrm{px}+\mathrm{q}=0\) was taken as 17 in place of 13 , then its roots were found to be \(-2\) and \(-15 .\) The roots of the original equation are (a) \(\\{-10,3\\}\) (b) \(\\{-10,-3\\}\) (c) \(\\{10,-3\\}\) (d) \(\quad\\{10,3\\}\)

Short answer

Answer: The roots of the original quadratic equation are -10 and -3.

Step-by-step solution

Roots of the incorrect equation

We are given that the roots of the incorrect equation are -2 and -15. Let's find the sum and product of these roots: Sum of roots (Incorrect equation) = -2 - 15 = -17 Product of roots (Incorrect equation) = -2 * -15 = 30

Determine the incorrect coefficient

We know that the coefficient of x (p) in the incorrect equation was taken as 17 in place of the correct value 13. So, the incorrect equation is: \(x^2 + 17x + q = 0\)

Determine q using product of roots (incorrect equation)

We know that the product of roots is equal to the constant term, q. In the incorrect equation, we have found the product of roots to be 30. Thus, we have: \(q = 30\)

Write down the original equation

Now, we know the correct value of p is 13. Using the correct value for p and the value of q from step 3, we can write down the original equation as: \(x^2 + 13x + 30 = 0\)

Determine the roots of the original equation

To find the roots of the original equation, we can factor the equation into: \((x + 10)(x + 3) = 0\) So, the roots of the original equation are -10 and -3. The correct answer is (b) \(\{-10, -3\}\).

Key concepts

Quadratic Equation Roots

In order to understand quadratic equation roots, let's start by recognizing that a quadratic equation is generally expressed in the standard form of \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( a eq 0 \). The solutions to this equation, known as the roots, are the values of \( x \) that satisfy the equation. The roots can be found by various methods, including factoring, completing the square, or using the quadratic formula.

In the context of the exercise provided, the roots of the quadratic equation were initially incorrectly determined due to a mistake in the coefficient of \( x \). Correcting this mistake is imperative to finding the true roots of the equation. The roots carry significant importance since they provide the points where the parabola, which represents the quadratic equation graphically, intersects the x-axis. In practical terms, these solutions can represent feasible answers in real-world problems, such as the dimensions of a field or the time at which an object reaches the ground when thrown.

Factoring Quadratic Equations

Factoring is one of the simplest methods employed to find the roots of a quadratic equation. It involves expressing the quadratic in the form of a product of its factors, usually two binomials. The process involves looking for two numbers that not only add up to the coefficient of \( x \), the term \( b \), but also multiply together to give the constant term \( c \).

In the provided exercise, factoring played a crucial role. Once the erroneous coefficient was corrected and the constant \( q \) was identified as the product of the roots (30 in this case), the equation was rewritten. By finding the correct factors, \( x^2 + 13x + 30 = 0 \) was factored down to \( (x + 10)(x + 3) = 0 \), which readily gives the roots as \( x = -10 \) and \( x = -3 \). This approach is highly useful and often quicker when the quadratic equation is factorable into integer root solutions.

Sum and Product of Roots

The sum and product of roots of a quadratic equation provide a direct path to understanding the nature of the equation without explicitly solving it. According to Vieta's formulas, for a quadratic equation in the form \( ax^2 + bx + c = 0 \), the sum of the roots \( (\alpha + \beta) \) is equal to \( -\frac{b}{a} \) and the product of the roots \( (\alpha \beta) \) is equal to \( \frac{c}{a} \). These relationships imply two things: no matter what the quadratic equation, if you know two of these values, you can find the third. This concept was intergral in solving our exercise.

Initially, in our exercise, the sum of the incorrect roots was determined to be -17 and the product was 30. By applying Vieta's formulas backwards, we can identify the constant term of the quadratic equation (\( q \)) and then use the correct coefficient for \( x \) to reconstruct and solve the original quadratic equation. Understanding these key relationships between the roots and the coefficients is fundamental when tackling a wide variety of problems involving quadratic equations, such as optimizing areas or modeling trajectories.