Problem 92 - Precalculus Student Solutions Manual 5th

Question

Show that the discriminant for the equation $$ \sqrt{2} x^{2}+5 x-3 \sqrt{2}=0 $$ is 49. If this equation is completely solved, it can be shown that the solution set is \(\left\\{-3 \sqrt{2}, \frac{\sqrt{2}}{2}\right\\} .\) We have a discriminant that is positive and a perfect square, yet the two solutions are irrational. Does this contradict the discussion in this section? Explain.

Step-by-Step Solution

Answer

The solutions \(-3\sqrt{2}\) and \(\frac{\sqrt{2}}{2}\) do not contradict the discussion because the coefficients are irrational.

Step 1: Identify the coefficients

For the quadratic equation of the form \(ax^2 + bx + c = 0\), identify the coefficients: \(a = \sqrt{2}\), \(b = 5\), and \(c = -3\sqrt{2}\).

Step 2: Calculate the discriminant

The discriminant \(\Delta\) of a quadratic equation \(ax^2 + bx + c = 0\) is given by the formula: \[\Delta = b^2 - 4ac\]Substituting the coefficients: \[\Delta = 5^2 - 4(\sqrt{2})(-3\sqrt{2})\]\[\Delta = 25 - 4(2)(-3)\]\[\Delta = 25 - (-24)\]\[\Delta = 25 + 24\]\[\Delta = 49\]

Step 3: Solve the quadratic equation

For the quadratic equation \(a x^2 + bx + c = 0\), the solutions can be found using the quadratic formula: \[x = \frac{-b \pm \sqrt{\Delta}}{2a}\]Substitute the values of \(a\), \(b\), and \(\Delta\): \[x = \frac{-5 \pm \sqrt{49}}{2\sqrt{2}}\]Note that \(\sqrt{49} = 7\): \[x = \frac{-5 \pm 7}{2\sqrt{2}}\]This gives two solutions: \[x_1 = \frac{-5 + 7}{2\sqrt{2}} = \frac{2}{2\sqrt{2}} = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}\]\[x_2 = \frac{-5 - 7}{2\sqrt{2}} = \frac{-12}{2\sqrt{2}} = \frac{-6}{\sqrt{2}} = -3\sqrt{2}\]

Step 4: Verify the discussion context

Usually, when the discriminant is a perfect square, the solutions are rational numbers. However, in this case, the coefficients contain irrational numbers (like \(\sqrt{2}\)), leading to irrational solutions. The fact that the solutions are irrational does not contradict the discussion, as the form of the coefficients influences the rationality of the solutions. Thus, there is no contradiction.

Key Concepts

Quadratic Formula

The quadratic formula is a vital tool for solving quadratic equations. It can solve any quadratic equation in the form of o(a x^2+bx+c=0) no unless the equation is impossible or inadequate. This formula allows you to find the values of x (the solutions) given the coefficients a, b, and c.

The formula is expressed as: o(x = \frac{-b \/pm \/sqrt{b^2-4ac}}{2a}). no Here, o(b^2-4ac) no is called the discriminant. By substituting a, b, and c into the formula, you can determine the roots of the quadratic equation.

Here's a step-by-step example: Given the equation o\( \sqrt{2}x^2+5x-3\sqrt{2}=0\),no the coefficients are o(a=\sqrt{2}), no (b=5), no and o(c=-3\sqrt{2}).no Plug them into the quadratic formula, calculate the discriminant, no(no\Delta=\sqrt{2}\ no), and find the solutions as: x=o(-5 \/pm 7)/(2\sqrt{2}).no This helps in understanding the power and application of the quadratic formula.

Irrational Solutions

Solutions to quadratic equations can sometimes be irrational, even when we expect rational ones. The discriminant helps us determine the nature of the solutions.

If the discriminant (o\Delta=b^2-4acno) is a positive perfect square, the solutions are usually rational. However, when the coefficients (a, b, or c) involve irrational numbers, the solutions can also be irrational even if o(\Delta)no is a perfect square.

In our practice equation, o\(\sqrt{2}x^2+5x-3\sqrt{2}=0\),no the discriminant is o\Delta=49.no Although the discriminant is a perfect square, the solutions are irrational, given by o(\{-3\sqrt{2}, \frac{\sqrt{2}}{2}\}).no This shows that the irrationality of the coefficients can lead to irrational solutions.

Perfect Square

A perfect square is a number obtained by squaring an integer. In the context of quadratic equations, if the discriminant is a perfect square, we often anticipate simple, rational solutions.

In our equation, o\(\sqrt{2}x^2+5x-3\sqrt{2}=0\),no the discriminant is calculated as: no(\Delta=49),no which is o7^2.no Normally, we'd expect rational solutions from a perfect square discriminant.

However, the coefficients of our equation contain the irrational number no(\sqrt{2}).no Here, the presence of the no\sqrt{2}),no in the coefficients influences the nature of solutions. As a result, even though no(\Delta = 49)no is a perfect square, the solutions are o{-3\sqrt{2}, \frac{\sqrt{2}}{2}no) which are irrational due to the coefficients' form.

This example emphasizes that perfect square discriminants do not guarantee rational solutions if the coefficients involve irrational numbers.