1-The roots of x2 - 6x + m - 3=0 are x1 and x2.If 2x1 - 3x2= 2 is given, find m?

2-Given the inequalities

x-2
______ < 0 and x2 - 4x - 5<0 , find the solution set.
x-4

x2 - 6x + m -3 = 0

x2 - 6x - 3 = -m
Is a start. Then use the Quadradic Formula to find the factors.

1- To find the value of "m" using the given information, we can use Vieta's formulas for the sum and product of roots of a quadratic equation.

First, let's rewrite the quadratic equation in the standard form: x^2 - 6x + (m - 3) = 0.

According to Vieta's formulas, the sum of the roots (x1 + x2) is equal to the negation of the coefficient of the linear term divided by the coefficient of the quadratic term. In this case, the sum of the roots is 6.

So, we have x1 + x2 = 6.

Next, we're given the equation 2x1 - 3x2 = 2. We can use substitution to express one root in terms of the other.

Rearranging the equation, we get 2x1 = 3x2 + 2.

Now, we can substitute this expression into the sum of the roots equation: 3x2 + 2 + x2 = 6.

Combining like terms, we have 4x2 + 2 = 6.

Then, subtracting 2 from both sides, we get 4x2 = 4.

Finally, dividing both sides by 4, we find x2 = 1.

Substituting this value back into the equation 2x1 - 3x2 = 2, we get 2x1 - 3(1) = 2.

Simplifying, we have 2x1 - 3 = 2.

Adding 3 to both sides, we find 2x1 = 5.

Dividing both sides by 2, we have x1 = 5/2 or x1 = 2.5.

Now, we have both x1 and x2, so we can substitute them into the original quadratic equation to find the value of "m":

(x - x1)(x - x2) = 0
(x - 2.5)(x - 1) = 0

Expanding this, we get x^2 - 3.5x + 2.5 = 0.

Comparing this with the original equation, we can see that m - 3 is equal to 2.5.

Therefore, m = 2.5 + 3 = 5.5.

2- Let's solve the inequalities one by one:

Inequality 1: (x - 2) / (x - 4) < 0

To find the solution set, we need to determine the intervals where the expression (x - 2) / (x - 4) is negative.

First, we identify the critical points by setting the numerator and denominator equal to zero:

x - 2 = 0 => x = 2
x - 4 = 0 => x = 4

These values divide the number line into three intervals: (-∞, 2), (2, 4), and (4, +∞).

For each interval, we can choose a test value to determine the sign of the expression.

- Let's test the interval (-∞, 2): Plugging in x = 0, we have (0 - 2) / (0 - 4) = 2 / -4 = -1/2, which is negative.

- Now, let's test the interval (2, 4): Plugging in x = 3, we have (3 - 2) / (3 - 4) = 1 / -1 = -1, which is negative.

- Lastly, let's test the interval (4, +∞): Plugging in x = 5, we have (5 - 2) / (5 - 4) = 3 / 1 = 3, which is positive.

Thus, the solution set for the first inequality is (-∞, 2) U (2, 4).

Inequality 2: x^2 - 4x - 5 < 0

To solve this inequality, we can factor the quadratic expression:

(x - 5)(x + 1) < 0

Now, we examine the critical points by setting each factor equal to zero:

x - 5 = 0 => x = 5
x + 1 = 0 => x = -1

These values divide the number line into three intervals: (-∞, -1), (-1, 5), and (5, +∞).

Choose a test value for each interval to determine the sign of the expression:

- For the interval (-∞, -1), let's use x = -2: (-2 - 5)(-2 + 1) = -7(-1) = 7, which is positive.

- For the interval (-1, 5), let's use x = 0: (0 - 5)(0 + 1) = -5(1) = -5, which is negative.

- For the interval (5, +∞), let's use x = 6: (6 - 5)(6 + 1) = 1(7) = 7, which is positive.

Hence, the solution set for the second inequality is (-∞, -1) U (5, +∞).

Combining the solution sets for both inequalities, we find:

(-∞, 2) U (2, 4) ∩ (-∞, -1) U (5, +∞)