Question

Given that \(4^{\sec ^{2} \alpha+1} \mathrm{x}^{2}-8 \mathrm{x}+4 \tan ^{2} \beta-8 \tan \beta+5=0\) has real solutions, then (a) \(\alpha=\mathrm{n} \pi, \beta=\mathrm{n} \pi+\frac{\pi}{4}\) and \(\mathrm{x}=\pm 1\) (b) \(\alpha=\frac{\mathrm{n} \pi}{4}, \beta=\mathrm{n} \pi\) and \(\mathrm{x}=\frac{1}{4}\) (c) \(\alpha=\mathrm{n} \pi, \beta=\mathrm{n} \pi+\frac{\pi}{4}\) and \(\mathrm{x}=\frac{1}{4}\) (d) there is no real solution for \(x\) if either \(\alpha=R-\\{n \pi\\}\) or \(\beta=R-\\{n \pi+\pi / 4\\}\)

Short answer

Answer: The quadratic equation has real solutions for x when \(\alpha = n \pi\) and \(\beta = n \pi + \frac{\pi}{4}\), where x = ±1 and n is an integer.

Step-by-step solution

Discriminant of a quadratic equation

For a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant, denoted as D, is given by: $$D = b^2 - 4ac$$ For the quadratic equation to have real solutions, the discriminant must be greater than or equal to zero: $$D \geq 0$$

Apply the discriminant criteria to the given equation

For our given equation, we need to identify the coefficients and use the discriminant: \(a = 4^{\sec ^{2} \alpha+1}\) \(b = -8\) \(c = 4 \tan ^{2} \beta - 8 \tan \beta + 5\) Now, we can find the discriminant and apply the criteria: $$D = (-8)^2 - 4 \cdot a \cdot c$$ $$D \geq 0$$

Simplify the inequality involving the discriminant

We can simplify the inequality as follows: $$64 - 16(4^{\sec ^{2} \alpha+1})(4 \tan ^{2} \beta - 8 \tan \beta + 5) \geq 0$$ $$4 - 4(4^{\sec ^{2} \alpha+1})(4 \tan ^{2} \beta - 8 \tan \beta + 5) \geq 0$$ $$1 - (4^{\sec ^{2} \alpha+1})(4 \tan ^{2} \beta - 8 \tan \beta + 5) \geq 0$$

Check given options for the required conditions

Now, we need to compare the given options with the simplified inequality to find the correct one: (a) \(\alpha = n \pi\) and \(\beta = n \pi + \frac{\pi}{4}\): \(\sec^2(\alpha) = 1\) and \(\tan(\beta) = 1\). This satisfies the inequality and results in x = ±1. (b) \(\alpha = \frac{n \pi}{4}\) and \(\beta = n \pi\): This would not lead to the true inequality as \(\sec^2(\alpha) \neq 1\) and \(\tan(\beta) \neq 1\). (c) \(\alpha = n \pi\) and \(\beta = n \pi + \frac{\pi}{4}\) and \(x = \frac{1}{4}\): This contradicts the value of x found in option (a). (d) Since there is a real solution for x in option (a), this option is incorrect.

Choose the correct option

Based on the analysis, we find that option (a) is correct: $$\alpha = n \pi, \beta = n \pi + \frac{\pi}{4} \textrm{, and } x = \pm 1$$

Key concepts

Discriminant

In quadratic equations, the discriminant is a central concept that helps determine the nature of the roots (or solutions) of the equation. For a quadratic equation of the form \(ax^2 + bx + c = 0\), the discriminant is calculated by the formula:

• \(D = b^2 - 4ac\)

The value of the discriminant \(D\) reveals critical information:

• If \(D > 0\), there are two distinct real solutions.
• If \(D = 0\), there is one real solution (a repeated root).
• If \(D < 0\), there are no real solutions, only complex ones.

For real solutions to exist, the discriminant must be non-negative (\(D \geq 0\)). This is why checking the discriminant is an essential step in solving quadratic equations. In this exercise, our task is to ensure the discriminant condition \(D \geq 0\) for real solutions is satisfied.

Trigonometric Identities

Trigonometric identities play a vital role in simplifying and solving equations involving trigonometric functions, such as sine, cosine, tangent, and secant. They help transform complex trigonometric expressions into simpler forms. Let's explore some relevant identities used in this context:

• Pythagorean Identity: \(\sec^2(\theta) = 1 + \tan^2(\theta)\)
• Addition Formulas: Useful for expressing angles as sums or differences. For example, \(\tan(a + b)\).

In the given problem, understanding how \(\sec^2(\theta)\) and \(\tan(\theta)\) behave with specific angle identities, such as \(\alpha = n\pi\) and \(\beta = n\pi + \frac{\pi}{4}\), provides clarity. Knowing that for such values, \(\sec(\alpha)\) equals 1 and \(\tan(\beta)\) equals 1 makes it easier to determine if the conditions of the quadratic equation align with real solutions.

Real Solutions

Determining if a quadratic equation has real solutions is crucial when solving mathematical problems. Real solutions occur when all parts of the equation yield real numbers upon solving. This boils down to ensuring:

• The discriminant \(D\) is non-negative.
• Any variables involved take values that are real numbers.

In this exercise, we worked with a condition for real solutions by ensuring the discriminant condition \(D \geq 0\) was met. Additionally, this involves confirming that specific values of \(\alpha\) and \(\beta\) lead to real numbers for provided trigonometric terms like \(\sec^2\) and \(\tan\). Thus, validating conditions such as \(\alpha = n\pi\) and \(\beta = n\pi + \frac{\pi}{4}\) which simplify trigonometric terms to known real values, ensures the quadratic terms can result in real solutions. By carefully analyzing these conditions, we identify specific scenarios where the quadratic equation indeed provides real values for \(x\), such as in option (a) with \(x = \pm 1\).