2𝑥2 + 12𝑥 + 14 = 𝑎(𝑥 + 𝑏)2 + 𝑐 , find 𝑎, 𝑏 and 𝑐

The equation you provided is in the form of a perfect square trinomial. The general form of a perfect square trinomial is a(x + b)^2 + c.

To find a, b, and c, we need to rewrite the given equation 2x^2 + 12x + 14 in the form of a perfect square trinomial.

Step 1: Divide the entire equation by the coefficient of x^2, which is 2. This gives us:

x^2 + 6x + 7 = a(x + b)^2 + c

Step 2: The coefficient of x in the simplified equation is 6. Half of this value is 3, so b = 3.

Step 3: Substitute b = 3 into the equation. This gives us:

x^2 + 6x + 7 = a(x + 3)^2 + c

Step 4: Expand the right side of the equation:

x^2 + 6x + 7 = a(x^2 + 6x + 9) + c

Step 5: To find a, we compare the coefficients of x^2 on both sides of the equation. The coefficient of x^2 on the left side is 1, and on the right side it is a. Therefore, a = 1.

Step 6: To find c, we compare the constant terms on both sides of the equation. The constant term on the left side is 7, and on the right side it is 9a + c. Substituting a = 1 gives us 7 = 9 + c. Solving for c gives us c = -2.

So, a = 1, b = 3, and c = -2.