Problem 21 - Algebra 1 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

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) \(-4\)

Step-by-Step Solution

Answer

The value of \(k\) is 0.

Step 1: Analyzing the Conditions for Roots

To have real and distinct roots, the discriminant of the quadratic equation must be greater than zero. For the expression \((k-2)x^2 + 8x + (k + 4) = 0\), the discriminant \(\Delta\) is given by \(b^2 - 4ac\). Here, \(a = k-2\), \(b = 8\), and \(c = k+4\). Therefore, \(\Delta = 8^2 - 4(k-2)(k+4) > 0\).

Step 2: Calculating the Discriminant

Calculate the discriminant: \[\Delta = 64 - 4((k-2)(k+4)) = 64 - 4(k^2 + 2k - 8)\]. Simplifying, \[\Delta = 64 - 4k^2 - 8k + 32\], which becomes \[96 - 4k^2 - 8k > 0\].

Step 3: Solving the Inequality

Simplify the inequality to find the suitable values of \(k\): \[-4k^2 - 8k + 96 > 0\]. Dividing by -4 gives \[k^2 + 2k - 24 < 0\]. Factorize the quadratic inequality: \[(k-4)(k+6) < 0\]. The solution implies \(-6 < k < 4\).

Step 4: Considering Negative Roots

For both roots to be negative, the condition \(-b/a > 0\) where \(a > 0\) must hold. Here, \(-b/a = -8/(k-2) > 0\). Thus \(k - 2 < 0\), which implies \(k < 2\).

Key Concepts

Real and Distinct Roots

When it comes to quadratic equations, having real and distinct roots is pivotal for various applications in mathematics. Real and distinct roots mean that the solutions to the equation are two different real numbers. This scenario occurs when the discriminant of the quadratic equation is greater than zero. The discriminant, symbolized by \( \Delta \), is calculated using the formula \( b^2 - 4ac \). Here, \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \). If \( \Delta > 0 \), the equation will have two distinct real solutions. It ensures that the roots are not equal and not complex, making them genuine real numbers.
Understanding when roots are real and distinct helps in solving systems of equations, modeling physical phenomena, and performing various mathematical computations with confidence.

Discriminant

The discriminant \( \Delta \) is a crucial component of the quadratic formula. It determines the nature of the roots of a quadratic equation \( ax^2 + bx + c = 0 \). The formula for the discriminant is \( \Delta = b^2 - 4ac \).

• If \( \Delta > 0 \), the equation has two real and distinct roots.
• If \( \Delta = 0 \), there is exactly one real root, or the roots are real and equal.
• If \( \Delta < 0 \), the roots are complex and conjugate pairs.

The discriminant doesn’t only indicate the number and type of roots, but also sheds light on the geometric aspect of the graph of the quadratic function. A positive discriminant depicts a parabola intersecting the x-axis at two distinct points, while a zero discriminant shows a parabola tangential to the x-axis at one point.

Inequality Solutions

Solving inequalities involving quadratic expressions requires careful analysis of the sign changes of the expression. For the quadratic \((k - 4)(k + 6) < 0\), the focus is on determining where the product of the two factors is negative. This condition occurs between the roots of the equation \( k^2 + 2k - 24 = 0 \), which are found by factorization or by solving \( (k-4)(k+6) = 0 \). Here, the roots are \( k = 4 \) and \( k = -6 \).
The inequality \((k - 4)(k + 6) < 0\) means that \( k \) must lie between \( -6 \) and \( 4 \). Using a number line can help in visualizing the intervals where the expression is negative. This involves testing values in the intervals determined by the roots to understand where the expression changes from negative to positive, resulting in valid solutions for \( k \).

Negative Roots

For a quadratic equation to yield negative roots, every solution obtained for \( x \) needs to be less than zero. This condition is often guided by both algebraic manipulations and logic. In the given equation, another condition to check is \(-b/a > 0\). This condition ensures that the average of the roots, which is represented by \(-b/a\), is positive when multiplied by the leading coefficient if it is flipped in sign.
Specifically, in our context, we require \(-8/(k-2) > 0\) to determine that both roots are negative. This simplifies to requiring that \( k - 2 < 0 \), which means \( k < 2 \). Only under such conditions will the quadratic equation yield roots that are entirely negative, fitting the constraints of the problem.
Understanding these conditions thoroughly helps to ensure solutions to equations are correct and conform to specified characteristics.