1: Solve the following exponential equation 4^(x)+2^(x+2)-12=0

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1: Solve the following exponential equation  4^(x)+2^(x+2)-12=0

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Identify Equation Structure: Let's first identify the structure of the equation and see if we can simplify it using properties of exponents. The equation is 4^(x) + 2^(x+2) - 12 = 0. We notice that 4 is a power of 2, specifically 4 = 2^2, so we can rewrite 4^(x) as (2^2)^(x) = 2^(2x). Also, 2^(x+2) can be written as 2^x * 2^2 = 4 * 2^x. Now, let's rewrite the equation with these observations.Rewrite Using Exponents: Rewrite the equation using the properties of exponents. 4^(x) + 2^(x+2) - 12 = 0 becomes 2^(2x) + 4 * 2^x - 12 = 0. Now, we have an equation that resembles a quadratic equation in form of 2^x. Let's set y = 2^x and rewrite the equation in terms of y.Substitute and Rewrite: Substitute y = 2^x into the equation. The equation becomes y^2 + 4y - 12 = 0. Now, we have a quadratic equation in y, which we can solve using factoring, completing the square, or the quadratic formula. Let's try to factor the quadratic equation.Factor Quadratic Equation: Factor the quadratic equation y^2 + 4y - 12 = 0. We look for two numbers that multiply to -12 and add up to 4. The numbers 6 and -2 satisfy these conditions. So, we can write the equation as (y + 6)(y - 2) = 0. Now, we can find the values of y by setting each factor equal to zero.Solve for y: Solve for y from the factors (y + 6) = 0 and (y - 2) = 0. For the first factor, y + 6 = 0, we get y = -6. For the second factor, y - 2 = 0, we get y = 2. Now we have two possible values for y, which are y = -6 and y = 2.Solve for x: Recall that y = 2^x. We will now solve for x using the values of y we found. First, let's take y = -6. Since 2^x is always positive, there is no real number x such that 2^x = -6. Therefore, y = -6 does not give us a valid solution for x. Now, let's take y = 2. We need to solve 2^x = 2.Solve the equation 2^x = 2. Since the bases are the same, we can equate the exponents. This gives us x = 1. So, the solution to the equation 2^x = 2 is x = 1.

$1$ : Solve the following exponential equation $\newline$ $4^{x}+2^{x+2}-12=0$

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$1$ : Solve the following exponential equation $\newline$ $4^{x}+2^{x+2}-12=0$