Question

If the difference of the roots of the equation \(x^{2}-p x+q=0\) is 1 then (a) \(p^{2}+4 q^{2}=(1+2 q)^{2}\) (b) \(\mathrm{q}^{2}+4 \mathrm{p}^{2}=(1+2 \mathrm{q})^{2}\) (c) \(p^{2}-4 q^{2}=(1+2 q)^{2}\) (d) \(q^{2}+4 p^{2}=(1-2 q)^{2}\)

Short answer

The correct answer is (a) \(p^2 + 4q^2 = (1 + 2q)^2\).

Step-by-step solution

Find roots using the quadratic formula

The quadratic formula states that the roots of a quadratic equation \(ax^2 + b*x + c = 0\) are: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) In our case, we have \(a=1, b=-p\), and \(c=q\). So the roots of the given quadratic equation are: \(x_1 = \frac{p + \sqrt{p^2 - 4q}}{2}\) and \(x_2 = \frac{p - \sqrt{p^2 - 4q}}{2}\)

Find the difference of the roots

The difference of the roots is: \(x_1 - x_2 = \frac{p + \sqrt{p^2 - 4q}}{2} - \frac{p - \sqrt{p^2 - 4q}}{2} = \sqrt{p^2 - 4q}\)

Use the given condition

According to the given condition, the difference of the roots is 1. Therefore: \(\sqrt{p^2 - 4q} = 1\) Squaring both sides, we get: \(p^2 - 4q = 1\), or \(4q = p^2 - 1\)

Check the given options

Now, we will go through each option to see which one is true: (a) \(p^2 + 4q^2 = (1 + 2q)^2\) Substituting \(4q = p^2 - 1\), we get: \((p^2 - 1) + 4q^2 = (1 + 2q)^2\) This equation holds true. (b) \(q^2 + 4p^2 = (1 + 2q)^2\) Substituting \(4q = p^2 - 1\), we get: \(q^2 + 4(p^2 - 1) = (1 + 2q)^2\) This equation does not hold true. (c) \(p^2 - 4q^2 = (1 + 2q)^2\) Substituting \(4q = p^2 - 1\), we get: \(1 - 4q^2 = (1 + 2q)^2\) This equation does not hold true. (d) \(q^2 + 4p^2 = (1 - 2q)^2\) Substituting \(4q = p^2 - 1\), we get: \(q^2 + 4(p^2 - 1) = (1 - 2q)^2\) This equation does not hold true. So, the correct answer is (a) \(p^2 + 4q^2 = (1 + 2q)^2\).

Key concepts

Quadratic Formula

When tackling quadratic equations, the quadratic formula is an invaluable tool for finding the roots efficiently. The general form of a quadratic equation is \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are known coefficients, and \(x\) represents the unknown variable. The quadratic formula reveals the solutions to this equation as:

\(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

This formula is derived from the process of completing the square in algebra and is universal for any quadratic equation. Its beauty lies in the \(\pm\) symbol, which indicates two possible values for \(x\), resulting in the two roots of the equation: \(x_1\) and \(x_2\). Using this tool simplifies the process and allows students to confidently solve for the roots of any quadratic equation.

Difference of Roots

Understanding the difference of the roots of a quadratic equation expands a student’s algebraic repertoire. Given two roots \(x_1\) and \(x_2\) found using the quadratic formula, their difference is simply \(x_1 - x_2\).

In the given exercise, the quadratic equation has the form \(x^2 - px + q = 0\), and after applying the quadratic formula, the roots are identified as:

\(x_1 = \frac{p + \sqrt{p^2 - 4q}}{2}\)
and
\(x_2 = \frac{p - \sqrt{p^2 - 4q}}{2}\)

When we compute the difference \(x_1 - x_2\), we notice that the value \(\frac{p}{2}\) cancels out, and we are left with \(\sqrt{p^2 - 4q}\). Matches between the exercise condition and algebraic findings are then sought by equating the calculated difference to the condition given (i.e., the difference is 1) and solving for the coefficients, which further underscores the interconnected nature of quadratic roots and coefficients.

Solving Quadratic Equations

Solving quadratic equations can sometimes feel like a daunting task, but with the right approach, students can find it to be quite straightforward. Once familiar with the quadratic formula, students can apply it to any equation of the form \(ax^2 + bx + c = 0\).

The exercise provided illustrates a practical application of the formula, where the solution process involves finding the roots of the equation, observing their differences, and then linking those differences to additional properties or conditions, like the stipulation of the roots’ difference being equal to 1.

This step-by-step procedure not only leads to the roots themselves but also to a deeper comprehension of the relationship between the quadratic coefficients \(a\), \(b\), and \(c\), and the roots' characteristics. As students practice these steps and recognize patterns, their proficiency in solving quadratic equations and their overall algebraic intuition will grow.