Determining Integral Values for a Positive Quadratic Expression

This solution explains how to find the number of integral values of m for which the given quadratic expression, $(10m-9)x^2 - 2mx + 1$, is always positive for all real numbers $x$. A quadratic expression $ax^2 + bx + c$ is always positive if and only if its leading coefficient ($a$) is positive and its discriminant ($\Delta$) is negative.

Conditions for a Quadratic Expression to be Always Positive

For the quadratic expression $ax^2 + bx + c$ to be always positive (i.e., $> 0$ for all $x \in \mathbb{R}$), two conditions must be met:

1. The coefficient of the $x^2$ term must be positive: $a > 0$. This ensures the parabola opens upwards.
2. The discriminant must be negative: $\Delta = b^2 - 4ac < 0$. This ensures the parabola does not intersect or touch the x-axis, meaning it stays entirely above the x-axis.

Applying Conditions to the Given Quadratic Expression

The given quadratic expression is $(10m-9)x^2 - 2mx + 1$.

Here, we identify the coefficients:

• $a = (10m-9)$
• $b = -2m$
• $c = 1$

Condition 1: Analyzing the Leading Coefficient ($a > 0$)

We need the coefficient of $x^2$ to be positive:

$$(10m-9) > 0$$

Adding 9 to both sides:

$$10m > 9$$

Dividing by 10:

$$m > \frac{9}{10}$$

So, the first condition requires $m$ to be greater than $0.9$.

Condition 2: Analyzing the Discriminant ($\Delta < 0$)

Next, we calculate the discriminant ($\Delta$) and set it to be less than zero:

$$\Delta = b^2 - 4ac$$

Substituting the coefficients:

$$\Delta = (-2m)^2 - 4(10m-9)(1)$$ $$\Delta = 4m^2 - 4(10m-9)$$ $$\Delta = 4m^2 - 40m + 36$$

Now, we apply the condition $\Delta < 0$:

$$4m^2 - 40m + 36 < 0$$

Divide the entire inequality by 4 to simplify:

$$m^2 - 10m + 9 < 0$$

To find the values of $m$ that satisfy this inequality, we first find the roots of the corresponding quadratic equation $m^2 - 10m + 9 = 0$. We can factor this equation:

$$(m-1)(m-9) = 0$$

The roots are $m=1$ and $m=9$.

Since the quadratic $m^2 - 10m + 9$ represents an upward-opening parabola, the expression is negative between its roots. Therefore, the inequality $m^2 - 10m + 9 < 0$ holds true when:

$$1 < m < 9$$

Combining Both Conditions for $m$

We need to find the values of $m$ that satisfy both conditions simultaneously:

• Condition 1: $m > 0.9$
• Condition 2: $1 < m < 9$

The intersection of these two ranges is $1 < m < 9$. This means $m$ must be strictly greater than 1 and strictly less than 9.

Finding the Number of Integral Values of $m$

The question asks for the number of integral values of $m$ in the range $1 < m < 9$.

The integers strictly between 1 and 9 are:

2, 3, 4, 5, 6, 7, 8

Counting these integers, we find there are 7 such values.

Conclusion

There are exactly 7 integral values of $m$ (namely 2, 3, 4, 5, 6, 7, and 8) for which the given quadratic expression $(10m-9)x^2 - 2mx + 1$ is always positive.