Problem 107 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

For what values of \(a\) does the function \(x^{2}+2(a-1) x+a+5\) has at least one positive root?

Step-by-Step Solution

Answer

The function \(f(x)=x^2+2(a-1)x+a+5\) has at least one positive root for the values of 'a' in the intervals \(a\in(-\infty, -1) \cup [4, \infty)\).

Step 1: Determine the discriminant of the quadratic function

To find the properties of the roots of a quadratic function, we first need to determine its discriminant (\(Δ\)). The discriminant is found using the formula \(Δ = b^2 - 4ac\) for a quadratic function of the form \(f(x) = ax^2 +bx + c\). For our given function, \(f(x) = x^2 + 2(a-1)x + a+5\), the coefficients are: - \(a = 1\) - \(b = 2(a-1)\) - \(c = a+5\) To find the discriminant: \( Δ = b^2 - 4ac \)

Step 2: Calculate the discriminant for the given function

Using the coefficients found in step 1, we plug them into the discriminant formula: \(Δ = [2(a-1)]^2 - 4(1)(a+5)\) Simplify the equation: \(Δ = 4(a-1)^2 - 4(a+5)\)

Step 3: Analyze when at least one positive root exists

For a quadratic function to have at least one positive root, the discriminant must be greater than or equal to zero (\(Δ\geq 0\)). So let's find the values of \(a\) that satisfy this inequality: \(4(a-1)^2 - 4(a+5) \geq 0\)

Step 4: Solve the inequality for 'a'

Now, we need to solve the inequality for the values of \(a\) that make the discriminant greater than or equal to zero: \(4(a-1)^2 - 4(a+5) \geq 0\) Divide both sides by 4: \((a-1)^2 - (a+5) \geq 0\) Expand and simplify: \(a^2 - 2a + 1 - a - 5 \geq 0\) Combining like terms: \(a^2 - 3a - 4 \geq 0\) We now have a quadratic inequality to solve for 'a': \(a^2 - 3a - 4 \geq 0\)

Step 5: Solve the quadratic inequality

To solve the quadratic inequality, we first find the roots when the equation is equal to zero, and then determine the sign of the inequality: \(a^2 - 3a - 4 = 0\) Factor the equation: \((a-4)(a+1) = 0\) So the roots are \(a=-1\) and \(a=4\). We can now use test intervals to find the signs of the inequality for different values of 'a'. Split the intervals as follows: - \(-\infty < a < -1\) - \(-1 \leq a < 4\) - \(4 \leq a < \infty\) Test using the factored equation \((a-4)(a+1)\): 1. For \(a<-1\) (e.g., \(a=-2\)): \((-6)(1) > 0\), so this interval satisfies the inequality. 2. For \(-1\leq a < 4\) (e.g., \(a=0\)): \((-4)(1)<0\), so this interval does not satisfy the inequality. 3. For \(4\leq a< \infty\) (e.g., \(a=5\)): \((1)(6)>0\), so this interval satisfies the inequality.

Step 6: Write the final solution

The intervals that satisfy the inequality are \(-\infty

Key Concepts

Discriminant of a Quadratic Function

The discriminant is a powerful concept that offers insights into the nature of roots for quadratic functions. Deriving from a quadratic function in the format of \(f(x) = ax^2 + bx + c\), the discriminant \(Δ\) is calculated with the formula \(Δ = b^2 - 4ac\). The discriminant tells us not just whether roots exist, but their nature. If \(Δ > 0\), there are two distinct real roots. When \(Δ = 0\), we have exactly one real root (also known as a repeated or double root). However, if \(Δ < 0\), the roots are complex and conjugate to each other.

To visualize why this is significant, you can imagine the curve of the quadratic function. If \(Δ > 0\), the curve touches the x-axis at two points. But when \(Δ = 0\), the curve only touches the x-axis at a single point, known as the vertex of the parabola. Finally, a negative discriminant means the curve does not intersect the x-axis at all. When solving for values of \(a\) that yield positive roots, we desire that \(Δ\) be non-negative to ensure real solutions.

Roots of Quadratic Equation

The roots of a quadratic equation can be found using the formula \(x = \frac{-b \pm \sqrt{Δ}}{2a}\), when \(Δ\) is the discriminant and \(a\), \(b\), and \(c\) are coefficients from the standard form \(ax^2+bx+c=0\). Roots are the x-values where the quadratic function intersects the x-axis, and are also known as zeros or solutions of the equation.

For the equation to have at least one positive root, at least one solution must be greater than zero. This is essential to ensure that the parabola represented by the quadratic function crosses the positive side of the x-axis. By evaluating the function and its discriminant, we are able to determine the range of values for \(a\) that lead to the desired nature of roots, which is crucial for defining specific characteristics of the function and for solving real-world problems that can be modeled by quadratic equations.

Quadratic Function Analysis

Analyzing a quadratic function extends beyond just finding the roots. It includes understanding the function's graph, which is a parabola. The vertex provides the maximum or minimum value of the function depending on whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

In addition to the discriminant, which cues us into the number and type of roots, other aspects like the axis of symmetry (\(x = -\frac{b}{2a}\)) and the y-intercept (\(c\), the constant term) further detail the function's characteristics. Conducting a thorough quadratic function analysis allows for predicting the function's behavior across different domains and is particularly useful in applications such as projectile motion, business profit models, and natural phenomena.

Solving Quadratic Inequalities

Solving quadratic inequalities is akin to solving quadratic equations but in this case, we seek not only the precise points of intersection (the roots) but also the intervals in which the inequality holds true. The process involves finding the roots of the associated quadratic equation and then determining the sign of the parabola's value within different intervals of the variable.

For instance, the inequality \(a^2 - 3a - 4 \geq 0\) can be approached by first solving the equation \(a^2 - 3a - 4 = 0\) and then testing the intervals on either side of these roots to see where the inequality is satisfied. The solution to a quadratic inequality is a range of values that create a true statement when substituted into the inequality; this range may be continuous or may consist of disjoint intervals. In real-life applications, such as determining the range of feasible solutions within constraints or finding the period during which a business is profitable, mastering the solving of quadratic inequalities is indispensable.

Previous

Problem 106

Next

Problem 108

Other exercises in this chapter

Problem 105

Find all the values of \(k\) for which one root of the function \(x^{2}-(k+1) x+k^{2}+k-8\) exceeds 2 and the other root is less than \(2 ?\)

View solution

Problem 106

For what values of \(k\), one root of the function \((k-5) x^{2}-2 k x+k-4\) is smaller than 1 and the other root exceeds 2?

View solution

Problem 108

Let \(a, b, c\) be real. If \(a x^{2}+b x+c=0\) has two real roots \(\alpha\) and \(\beta\), where \(\alpha1\), then show that \(1+\frac{c}{a}+\left|\frac{b}{a}

View solution

Problem 109

Prove that the value of \(a\) for which \(2 x^{2}-2(2 a+1) x+a(a+1)=0\) may have one root less than \(a\) and the other root greater than \(a\), are given by \(

View solution

Key Concepts

Discriminant of a Quadratic Function

The discriminant is a powerful concept that offers insights into the nature of roots for quadratic functions. Deriving from a quadratic function in the format of \(f(x) = ax^2 + bx + c\), the discriminant \(Δ\) is calculated with the formula \(Δ = b^2 - 4ac\). The discriminant tells us not just whether roots exist, but their nature. If \(Δ > 0\), there are two distinct real roots. When \(Δ = 0\), we have exactly one real root (also known as a repeated or double root). However, if \(Δ < 0\), the roots are complex and conjugate to each other.

To visualize why this is significant, you can imagine the curve of the quadratic function. If \(Δ > 0\), the curve touches the x-axis at two points. But when \(Δ = 0\), the curve only touches the x-axis at a single point, known as the vertex of the parabola. Finally, a negative discriminant means the curve does not intersect the x-axis at all. When solving for values of \(a\) that yield positive roots, we desire that \(Δ\) be non-negative to ensure real solutions.

Roots of Quadratic Equation

The roots of a quadratic equation can be found using the formula \(x = \frac{-b \pm \sqrt{Δ}}{2a}\), when \(Δ\) is the discriminant and \(a\), \(b\), and \(c\) are coefficients from the standard form \(ax^2+bx+c=0\). Roots are the x-values where the quadratic function intersects the x-axis, and are also known as zeros or solutions of the equation.

For the equation to have at least one positive root, at least one solution must be greater than zero. This is essential to ensure that the parabola represented by the quadratic function crosses the positive side of the x-axis. By evaluating the function and its discriminant, we are able to determine the range of values for \(a\) that lead to the desired nature of roots, which is crucial for defining specific characteristics of the function and for solving real-world problems that can be modeled by quadratic equations.

Quadratic Function Analysis

Analyzing a quadratic function extends beyond just finding the roots. It includes understanding the function's graph, which is a parabola. The vertex provides the maximum or minimum value of the function depending on whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

In addition to the discriminant, which cues us into the number and type of roots, other aspects like the axis of symmetry (\(x = -\frac{b}{2a}\)) and the y-intercept (\(c\), the constant term) further detail the function's characteristics. Conducting a thorough quadratic function analysis allows for predicting the function's behavior across different domains and is particularly useful in applications such as projectile motion, business profit models, and natural phenomena.

Solving Quadratic Inequalities

Solving quadratic inequalities is akin to solving quadratic equations but in this case, we seek not only the precise points of intersection (the roots) but also the intervals in which the inequality holds true. The process involves finding the roots of the associated quadratic equation and then determining the sign of the parabola's value within different intervals of the variable.

For instance, the inequality \(a^2 - 3a - 4 \geq 0\) can be approached by first solving the equation \(a^2 - 3a - 4 = 0\) and then testing the intervals on either side of these roots to see where the inequality is satisfied. The solution to a quadratic inequality is a range of values that create a true statement when substituted into the inequality; this range may be continuous or may consist of disjoint intervals. In real-life applications, such as determining the range of feasible solutions within constraints or finding the period during which a business is profitable, mastering the solving of quadratic inequalities is indispensable.