Find the maximum integer value of m for which the quadratic inequality x^2 - mx + 9 > 0 ho… — JAMB UTME

Find the maximum integer value of $m$ for which the quadratic inequality $x^2 - mx + 9 > 0$ holds for all real values of $x$ .

Answer: 5

The maximum integer value of

m

5

For the quadratic expression

x^2 - mx + 9

to remain strictly positive for all real values of

x

, the quadratic curve must lie completely above the x-axis. Because the coefficient of

x^2

is positive (

1 > 0

), this requires the discriminant to be strictly negative (

D < 0

). Evaluating

b^2 - 4ac < 0

gives

m^2 - 36 < 0

, which simplifies to

-6 < m < 6

. The largest integer strictly less than

6

5

Determine the condition for the quadratic expression to be positive for all real values of

x

Since the leading coefficient is

1 > 0

, the condition is that the discriminant

D = b^2 - 4ac < 0

A parabola opening upward lies entirely above the horizontal axis when it has no real roots.

Calculate the discriminant using the coefficients of the quadratic expression.

D = (-m)^2 - 4(1)(9) = m^2 - 36 < 0

Here

a = 1

b = -m

, and

c = 9

Solve the quadratic inequality for

m

m^2 - 36 < 0 \implies -6 < m < 6

The roots of

m^2 - 36 = 0

are

m = -6

and

m = 6

, and the expression is negative strictly between these boundary values.

Determine the maximum integer value within the open interval

(-6, 6)

The maximum integer value is

5

The boundary value

6

is excluded by the strict inequality

m < 6

Quadratic Inequalities and Discriminant Conditions for Positive Definiteness

Question