Question

Solve each exponential equation. Express irrational solutions in exact form. $$ 2^{2 x}+2^{x+2}-12=0 $$

Short answer

x = 1

Step-by-step solution

- Write in a common base

Observe that both terms on the left side of the equation can be written with base 2. Rewrite the equation as follows: \[2^{2x} + 2^{x+2} - 12 = 0\]This step helps to recognize patterns in the exponents.

- Simplify the exponentials

Use properties of exponents to simplify the equation:\[2^{2x} = (2^x)^2\] and \[2^{x+2} = 2^x \times 2^2 = 4 \times 2^x\]So, the equation becomes:\[(2^x)^2 + 4 \times 2^x - 12 = 0\]

- Substitute with a new variable

Let \(y = 2^x\). Substitute this into the equation to get a quadratic form:\[y^2 + 4y - 12 = 0\]

- Solve the quadratic equation

Solve the quadratic equation using the quadratic formula \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). For \(a=1, b=4, c=-12\):\[y = \frac{-4 \pm \sqrt{16 + 48}}{2}\] \[y = \frac{-4 \pm 8}{2}\]This yields two solutions:\[y = 2\] and \[y = -6\]Since \(2^x > 0\), the valid solution is \(y = 2\).

- Back-substitute and solve

Recall that \(y = 2^x\). So, \[2^x = 2\] yields:\[x = 1\].

Key concepts

properties of exponents

Exponents are fundamental in solving exponential equations. Understanding their properties is key.
An exponent, or power, indicates how many times a number, called the base, is multiplied by itself. For example, in the expression \(2^3\), 3 is the exponent, and 2 is the base, making the expression equal to 8.
Key properties include:

• Product of Powers: \(a^m \times a^n = a^{m+n}\)
• Quotient of Powers: \(a^m / a^n = a^{m-n}\)
• Power of a Power: \((a^m)^n = a^{m \times n}\)
• Power of a Product: \((ab)^n = a^n \times b^n \)
• Zero Exponent: \(a^0 = 1\) (assuming \(a eq 0\))
• Negative Exponent: \(a^{-n} = 1/a^n\) (again, assuming \(a eq 0\))

Understanding these properties allows for the simplification of complex exponential expressions, which is critical in solving equations like the one provided in the exercise.

quadratic equations

Quadratic equations are another essential tool in solving exponential equations. A quadratic equation takes the standard form \(ax^2 + bx + c = 0\).
The quadratic formula \(x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a}\) provides solutions to any quadratic equation.
To apply the quadratic formula:

• Identify coefficients \(a, b,\) and \(c\).
• Calculate the discriminant \(b^2 - 4ac\).
• Substitute values into the quadratic formula.

In our exercise, substituting \(2^x = y\) converts the problem into the quadratic form \(y^2 + 4y - 12 = 0\).
The solutions tell us the possible values for \(y\), which we then interpret in terms of the original variable \(x\).

substitution method

The substitution method is a powerful technique for simplifying equations. It involves replacing a complex expression with a single variable to make solving more manageable.
For instance, in the exercise, substituting \(2^x\) with a simpler variable \(y\) transforms the exponential equation into a quadratic form. This makes the equation easier to solve.
To use the substitution method:

• Choose a substitution that simplifies the equation. For example, set \(y = 2^x\).
• Replace the original expression with the new variable.
• Solve the simpler equation.
• Back-substitute to return to the original variables.

In essence, substitution breaks down the problem into more manageable parts, easing the path to the solution.

exponential functions

Exponential functions are crucial in many mathematical and real-world contexts. An exponential function is of the form \(f(x) = a \, \cdot \, b^x\), where \(a\) and \(b\) are constants and \(b > 0\), \(b eq 1\).
These functions show exponential growth or decay.
Key components of exponential functions include:

• Base (\(b\)): Determines the rate of growth or decay. If \(b > 1\), the function grows. If \(0 < b < 1\), it decays.
• Exponent (\(x\)): The variable that changes the value of the function.

In solving exponential equations, converting terms to a common base often simplifies the problem. Like in the exercise, expressing everything in terms of base 2 made it easier to apply the properties of exponents.
Understanding exponential functions also helps in interpreting solutions in real-world scenarios where exponential growth or decay occurs.