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Find the range of values of m such that the equation in terms of variable x.

mx² + (2m-3)x+ (m-2) = 0

(a) has two equal real roots; (Ans: m =)

(b) has two distinct real roots; (Ans: m <-,m 0)

(c) has no real root. (Ans: m>)

2.If the equation in terms of variable x. 2x²- 7x + m= 0, has real roots, find the maximum integervalue of m. (Ans: 6 )

3.If the equation in terms of variable x, kx²+ (4k + 3)x + 4k - 2= 0, has two distinct real roots, find the range of values of k. (Ans: k>- k0)

4.If the equation in terms of variable x, 2kx² - 3x + 5= 0, has no real root, find the range of valuesof k. (Ans: k>)

Given that (3a + 4)x²- 4ax + 4 is a perfect square expression, find the value(s) of a. (Ans: -1 or4)

Given that a 0, prove that the equation in terms of x, a²x²+ 2ax + 2 = 0, has no real solutions.

Given that the equation in terms of variable x, a(1- x²) + 2bx + c(1+ x²) = 0, has equal roots,prove that c² = a² + b² and a

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ELSABABA08EN

1. For the equation mx² + (2m-3)x + (m-2) = 0:

(a) If it has two equal real roots, the discriminant should be 0, so (2m-3)² - 4m(m-2) = 0. Solve for m.

(b) For two distinct real roots, the discriminant should be greater than 0, so (2m-3)² - 4m(m-2) > 0. Find the range for m.

(c) If it has no real root, the discriminant should be negative, so (2m-3)² - 4m(m-2) < 0. Determine the range for m.

2. For the equation 2x² - 7x + m = 0 to have real roots, the discriminant (b² - 4ac) should be greater than or equal to 0. Find the maximum integer value of m.

3. For the equation kx² + (4k + 3)x + 4k - 2 = 0 to have two distinct real roots, the discriminant should be greater than 0. Solve for the range of values of k.

4. For the equation 2kx² - 3x + 5 = 0 to have no real root, the discriminant should be negative. Determine the range of values for k.

5. Given (3a + 4)x² - 4ax + 4 as a perfect square expression, the discriminant should be 0. Solve for the value(s) of a.

6. To prove that a²x² + 2ax + 2 = 0 has no real solutions when a > 0, you can use the discriminant and show that it's negative.

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