Question

Please help!! ASAP

1) Prove that the equation 7x^(2)+bx−23=0 for any value of b has one positive and one negative root.

2) Prove that the equation 12x^(2)+70x+a2+1=0 for any value of a does not have any positive roots.

3) Prove that for any natural n the value of the expression (4n+5)^(2)–9 is divisible by 4.

Answer 1

The equation 7x^2 + bx - 23 = 0 has one positive and one negative root for any value of b.

Step-by-step explanation:

To prove that the equation 7x^2 + bx - 23 = 0 has one positive and one negative root for any value of b, we can use the discriminant. The discriminant is the part of the quadratic formula that determines the number of roots. It is given by the formula: D = b^2 - 4ac. In this case, a = 7, b = b, and c = -23.

If the discriminant is positive, the equation has two distinct real roots. If it is zero, the equation has one real root. And if it is negative, the equation has no real roots.

Since the discriminant is b^2 - 4ac for this equation, we can see that it will always be positive. Therefore, the equation 7x^2 + bx - 23 = 0 has one positive and one negative root for any value of b.

Answer 2

#1 Picture above
#2 94x + a
#3 32n^2 + 80n + 41

They are in order by the question