Question

The integer \(\mathrm{k}\) for which the inequality \(\mathrm{x}^{2}-2(4 \mathrm{k}-1) \mathrm{x}+15 \mathrm{k}^{2}-2 \mathrm{k}-7>0\) is valid for real \(\mathrm{x}\) is (a) \(\overline{2}\) (b) 3 (c) 4 (d) 5

Short answer

a) 2 b) 3 c) 4 d) 5 Solution: Following the step-by-step analysis, we found that the inequality holds true for k = 3, 4, and 5. However, there might be an error in the question or it needs to be rephrased to include additional constraints or ask for all values of k that satisfy the inequality. Considering the given options, the answers could be (b) 3, (c) 4, or (d) 5. Recommending to review the question for possible errors.

Step-by-step solution

Rewrite the inequality

Rewrite the given inequality in standard quadratic form, which is: $$ax^2 + bx + c > 0$$ The given inequality is: $$x^2 - 2(4k - 1)x + 15k^2 - 2k - 7 > 0$$

Find the discriminant

The discriminant (Δ) of a quadratic function is given by: $$Δ = b^2 - 4ac$$ In our case: $$a = 1$$ $$b = -2(4k - 1)$$ $$c = 15k^2 - 2k - 7$$ Calculate the discriminant: $$Δ = (-2(4k - 1))^2 - 4(1)(15k^2 - 2k - 7)$$

Analyze the discriminant

For the inequality to hold for real values of x, the discriminant (Δ) must be greater than zero: $$Δ > 0$$ Substitute the discriminant we found in step 2: $$(-2(4k - 1))^2 - 4(1)(15k^2 - 2k - 7) > 0$$

Simplify and solve for k

Simplify the inequality: $$16(4k - 1)^2 - 60k^2 + 8k + 28 > 0$$ Now we need to solve for k: We will test each given option for k to check which one satisfies the inequality.

Test options (a) 2, (b) 3, (c) 4, (d) 5

For (a) \(\overline{2}\), testing integers is not appropriate. So we can eliminate option (a). Test option (b) k = 3: $$16(4(3) - 1)^2 - 60(3)^2 + 8(3) + 28 > 0$$ $$16(11)^2 - 60(9) + 24 + 28 > 0$$ $$1936 - 540 + 52 > 0$$ $$1448 > 0$$ The inequality holds true for k = 3. Test option (c) k = 4: $$16(4(4) - 1)^2 - 60(4)^2 + 8(4) + 28 > 0$$ $$16(15)^2 - 60(16) + 32 + 28 > 0$$ $$3600 - 960 + 60 > 0$$ $$2700 > 0$$ The inequality holds true for k = 4. Test option (d) k = 5: $$16(4(5) - 1)^2 - 60(5)^2 + 8(5) + 28 > 0$$ $$16(19)^2 - 60(25) + 40 + 28 > 0$$ $$5776 - 1500 + 68 > 0$$ $$4344 > 0$$ The inequality holds true for k = 5. According to our tests, the inequality holds true for k = 3, 4, and 5. However, the question asks for the integer k, which implies that there should be only one correct answer. This means there might be an error in the question, or it needs to be rephrased to include additional constraints or ask for all values of k that satisfy the inequality. Considering the given options, the answers could be (b) 3, (c) 4, or (d) 5.

Key concepts

Discriminant of Quadratic Equation

Understanding the discriminant is crucial when dealing with quadratic equations. In essence, the discriminant reveals the nature of the roots of a quadratic equation, which is of the form \( ax^2 + bx + c = 0 \). The discriminant, denoted as \( \Delta \), is calculated using the formula \( \Delta = b^2 - 4ac \).

Here's why the discriminant is so important:

• Positive Discriminant (\(\Delta > 0\)): Implies two distinct real roots.
• Zero Discriminant (\(\Delta = 0\)): Leads to one real, repeated root (also known as a double root).
• Negative Discriminant (\(\Delta < 0\)): Indicates complex roots with no real solutions.

The exercise provided involves a quadratic inequality rather than an equation. However, understanding the discriminant is equally essential as it determines if real values of \( x \) exist that can satisfy the inequality.

Solving Quadratic Inequalities

Solving quadratic inequalities such as \( ax^2 + bx + c > 0 \) involves several steps but the process is similar to solving quadratic equations. The aim is to find the range of values of \( x \) for which the inequality holds true. Here are simplified steps for solving quadratic inequalities:

• Rewrite the inequality in the standard form.
• Find the discriminant to determine the nature of the roots.
• Analyze the roots based on the discriminant:
• If the discriminant is positive, find the real roots and determine the intervals on which the inequality is satisfied.
• If the discriminant is zero or negative, this indicates either one real root or no real roots respectively, affecting the set of solutions for \( x \).
• Test values from each interval to see which satisfy the inequality.

Solving these inequalities not only involves calculation but also interpretation of results to understand where the inequality is valid.

Real Values of x

When solving quadratic inequalities, the solution set often requires identifying the real values of \( x \) that satisfy the condition. These real values can be single points, intervals, or a combination thereof, depending on the inequality given.

In the exercise, we seek a value of \( k \) for which the quadratic inequality holds true for all real \( x \) values. The process reveals that certain values of \( k \) produce a positive discriminant which hints at real solutions and intervals where the inequality is valid.

Identifying these values involves testing the inequality at different intervals determined by the found roots, if any. Checking these intervals helps us determine where the function \( ax^2 + bx + c \) is positive or negative, thus finding the range of real \( x \) values that are solutions to the inequality.