Question

The value of \(p\) for which the difference between the roots of the equation \(x^{2}+p x+8=0\) is 2 are : (a) \(\pm 2\) (b) \(\pm 4\) (c) \(\pm 6\) (d) \(\pm 8\)

Short answer

Answer: (c) \(\pm 6\)

Step-by-step solution

Expressing the roots using the quadratic formula

The quadratic formula is given by \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\), where a, b, and c are the coefficients of the quadratic equation \(ax^{2}+bx+c=0\). In this case, \(a=1\), \(b=p\), and \(c=8\). So, the roots are \(x_{1,2}=\frac{-p\pm\sqrt{p^2-4(1)(8)}}{2(1)}\).

Finding the difference between the roots

The difference between the roots is given by $|x_1 - x_2| = \left|\frac{-p+\sqrt{p^2-32}}{2}-\frac{-p-\sqrt{p^2-32}}{2}\right|= \left|\frac{2\sqrt{p^2-32}}{2}\right|=\sqrt{p^2-32}$.

Setting up the equation and solving for p

We want the difference between the roots to be 2, so we have \(\sqrt{p^2-32}=2\). Squaring both sides, we get \(p^2-32=4\). This gives \(p^2=36\), and hence, \(p=\pm 6\). Therefore, the value of \(p\) for which the difference between the roots of the equation \(x^{2}+p x+8=0\) is 2 is (c) \(\pm 6\).

Key concepts

Quadratic Formula

The quadratic formula is a powerful tool used for solving quadratic equations, which are polynomial equations of the second degree in the form \(ax^{2}+bx+c=0\), where \(a\), \(b\), and \(c\) are known values and \(a \eq 0\). The formula is \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\). This formula makes it possible to find solutions for any quadratic equation by substituting the coefficients \(a\), \(b\), and \(c\) into the formula.

It should be noted that the discriminant \(\Delta=b^2-4ac\) inside the square root in the quadratic formula indicates the nature of the roots. If \(\Delta>0\), the equation has two distinct real roots. If \(\Delta=0\), it has exactly one real root (also called a repeated or double root). Finally, if \(\Delta<0\), the equation has no real roots but instead has two complex roots.

Difference Between Roots

The difference between the roots of a quadratic equation can be obtained without actually computing the roots themselves. If the roots of \(ax^{2} + bx + c = 0\) are \(x_1\) and \(x_2\), the absolute difference is given by \(\left| x_1 - x_2 \right|\), which simplifies to \(\left| \frac{\sqrt{b^2-4ac}}{a} \right|\) when derived from the quadratic formula.

This measure is essential in many real-world problems where the relative spacing of values is important. For instance, in physics, it could represent the difference in energy levels, and in economics, the gap between cost estimations.

Solving Quadratic Equations

Solving quadratic equations involves finding the values of \(x\) that satisfy the equation \(ax^{2}+bx+c=0\). There are various methods to do this, including factoring, completing the square, using the quadratic formula, and graphing. The quadratic formula is often the most direct method, especially when the equation does not factor easily.

It's important to follow a systematic approach: firstly, identify the coefficients and constant term, then use these values in the quadratic formula to find the roots, and finally, interpret the roots in the context of the problem to determine the most appropriate solution.

Coefficients of Quadratic Equation

The coefficients of a quadratic equation \(ax^{2}+bx+c=0\) are the numerical factors \(a\), \(b\), and \(c\). The coefficient \(a\) is tied to the curvature of the parabola represented by the equation, \(b\) controls the direction of the opening along with \(a\), and \(c\) represents the \(y\)-intercept of the parabola.

Understanding these coefficients is crucial for analyzing the motion of projectiles in physics, optimizing profit in business, or determining the maximum area in geometry. Each coefficient's value influences the location and shape of the parabola, affecting the problem's solution.