Problem 23 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The values of \(a\) for which one root of the equation \(x^{2}-(a+1) x+a^{2}+a-8=0\) exceeds 2 and the other is lesser than 2 , are given by (a) \(a>3\) (b) \(9

Step-by-Step Solution

Answer

Option (c) \(-2

Step 1: Identify the roots condition

For a quadratic equation with roots \(\alpha\) and \(\beta\), given that one root exceeds 2 and the other root is less than 2, implies \(\alpha > 2\) and \(\beta < 2\) or \(\alpha < 2\) and \(\beta > 2\). This means the value 2 is between the two roots.

Step 2: Express the roots in terms of coefficients

For the quadratic equation \(x^{2}-(a+1)x+a^{2}+a-8=0\), apply Vieta's formulas: the sum of the roots \(\alpha + \beta = a + 1\) and the product of the roots \(\alpha \beta = a^{2} + a - 8\).

Step 3: Develop inequality using critical points

Since 2 lies between the roots, substitute \(x = 2\) in the quadratic equation, so \((2)^{2} - (a+1)(2) + (a^{2} + a - 8) < 0\). Simplifying this gives \(4 - 2a - 2 + a^{2} + a - 8 < 0\) or \(a^{2} - a - 6 < 0\).

Step 4: Solve the inequality

Factor the quadratic inequality: \(a^{2} - a - 6 = (a - 3)(a + 2) < 0\). The critical points are \(-2\) and \(3\). Use the test point method or consider sign changes over intervals given by these critical points: \(-\infty < a < -2\), \(-2 < a < 3\), and \(3 < a < \infty\). The inequality holds in the interval \(-2 < a < 3\).

Step 5: Confirm the correct option

Review the options: (a) \(a>3\), (b) \(9

Key Concepts

Vieta's Formulas

Vieta's formulas are a powerful mathematical tool that relates the coefficients of a polynomial to sums and products of its roots. For any quadratic equation of the form \[x^2 + bx + c = 0 \] the sum and product of the roots can be directly derived from the coefficients as follows:

• The sum of the roots, denoted usually by \( \alpha + \beta \), is equal to \(-b\). So, for the equation given in the exercise, the sum is \(a+1\).
• The product of the roots, denoted by \( \alpha \beta \), is \(c\). Hence, in the equation given, the product of the roots is \(a^2 + a - 8\).

These relationships enable one to find the characteristics of the roots without explicitly solving the quadratic equation by factoring or using the quadratic formula. It means you can understand properties like the position of roots concerning a number line, which is especially useful in inequalities.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically having the form \[ax^2 + bx + c = 0.\] The solutions to these equations, called 'roots', can be found using factoring, completing the square, or applying the quadratic formula.In the given exercise, the specific equation \[x^2 - (a+1)x + a^2 + a - 8 = 0.\] is tackled by assessing the distribution of its roots concerning the point \(x=2\). Knowing that a certain critical value lies between two roots reveals the nature of the inequalities formed by the quadratic expression.Solving involves:

• Identifying the sum and product of the roots using Vieta's formulas.
• Factoring the left-hand side of the inequality formed.
• Using methods like testing intervals or substituting critical points to evaluate where the inequality holds.

These techniques help deduce whether an area on the number line satisfies the inequality condition derived from the quadratic.

Critical Points Analysis

Critical points analysis in the context of quadratic inequalities involves finding the locations on a number line where the expression equals zero, as these points separate areas of positive and negative value for the expression.For the inequality derived in the exercise: \[a^2 - a - 6 < 0,\] we first factor it as \[(a - 3)(a + 2) < 0.\]The critical points here are \(-2\) and \(3\). After determining these, the number line is divided into three intervals:

• \(-\infty < a < -2\)
• \(-2 < a < 3 \)
• \(3 < a < \infty \)

Evaluate the sign of each factor in the intervals to see where the expression satisfies the inequality. For \[(a - 3)(a + 2) < 0,\] the expression is negative between \(-2\) and \(3\), indicating its validity. This check ensures the solution fits the required conditions like having roots on either side of \(x=2\), informing us about possible values for 'a' in the original problem.

Previous

Problem 22

Next

Problem 24

Other exercises in this chapter

Problem 21

The value of \(k\) for which the equation \((k-2) x^{2}+8 x+k+4=0\) has both roots real, distinct and negative is (a) 0 (b) 2 (c) 3 (d) \(-4\)

View solution

Problem 22

The set of values of \(\lambda\) for which the equation \(3 x^{2}+2 x+\lambda(\lambda-1)=0\) are of opposite signs is (a) \((0,1)\) (b) \([0,1]\) (c) \([0,1)\)

View solution

Problem 24

The value of \(p\) for which both the roots of the equation \(4 x^{2}-20 p x+\left(25 p^{2}+15 p-66\right)\) \(=0\) are less than 2 , lies in the interval (a) \

View solution

Problem 25

If both the roots of \(a x^{2}+b x+c=0\) are positive, then (a) \(-\frac{b}{a}>0\) (b) \(\frac{c}{a}>0\) (c) \(b^{2} \geq 4 a c\) (d) \(a c>0\)

View solution

Key Concepts

Vieta's Formulas

Vieta's formulas are a powerful mathematical tool that relates the coefficients of a polynomial to sums and products of its roots. For any quadratic equation of the form \[x^2 + bx + c = 0 \] the sum and product of the roots can be directly derived from the coefficients as follows:

• The sum of the roots, denoted usually by \( \alpha + \beta \), is equal to \(-b\). So, for the equation given in the exercise, the sum is \(a+1\).
• The product of the roots, denoted by \( \alpha \beta \), is \(c\). Hence, in the equation given, the product of the roots is \(a^2 + a - 8\).

These relationships enable one to find the characteristics of the roots without explicitly solving the quadratic equation by factoring or using the quadratic formula. It means you can understand properties like the position of roots concerning a number line, which is especially useful in inequalities.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, typically having the form \[ax^2 + bx + c = 0.\] The solutions to these equations, called 'roots', can be found using factoring, completing the square, or applying the quadratic formula.In the given exercise, the specific equation \[x^2 - (a+1)x + a^2 + a - 8 = 0.\] is tackled by assessing the distribution of its roots concerning the point \(x=2\). Knowing that a certain critical value lies between two roots reveals the nature of the inequalities formed by the quadratic expression.Solving involves:

• Identifying the sum and product of the roots using Vieta's formulas.
• Factoring the left-hand side of the inequality formed.
• Using methods like testing intervals or substituting critical points to evaluate where the inequality holds.

These techniques help deduce whether an area on the number line satisfies the inequality condition derived from the quadratic.

Critical Points Analysis

Critical points analysis in the context of quadratic inequalities involves finding the locations on a number line where the expression equals zero, as these points separate areas of positive and negative value for the expression.For the inequality derived in the exercise: \[a^2 - a - 6 < 0,\] we first factor it as \[(a - 3)(a + 2) < 0.\]The critical points here are \(-2\) and \(3\). After determining these, the number line is divided into three intervals:

• \(-\infty < a < -2\)
• \(-2 < a < 3 \)
• \(3 < a < \infty \)

Evaluate the sign of each factor in the intervals to see where the expression satisfies the inequality. For \[(a - 3)(a + 2) < 0,\] the expression is negative between \(-2\) and \(3\), indicating its validity. This check ensures the solution fits the required conditions like having roots on either side of \(x=2\), informing us about possible values for 'a' in the original problem.