Question

Solve each exponential equation. Express irrational solutions in exact form. $$ 3 \cdot 4^{x}+4 \cdot 2^{x}+8=0 $$

Short answer

The equation has no real solutions.

Step-by-step solution

Simplify the equation

Start by rewriting the terms to make the bases consistent. Express all terms with base 2: \(4^x = (2^2)^x = 2^{2x}\) and \(8 = 2^3\). Now, the equation can be re-written as: \(3 \times 2^{2x} + 4 \times 2^x + 2^3 = 0\).

Introduce a substitution

Let \(y = 2^x\). The equation then becomes: \(3y^2 + 4y + 8 = 0\).

Solve the quadratic equation

Use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \(a = 3\), \(b = 4\), and \(c = 8\): \[ y = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 3 \cdot 8}}{2 \cdot 3} = \frac{-4 \pm \sqrt{16 - 96}}{6}\] \[ y = \frac{-4 \pm \sqrt{-80}}{6} = \frac{-4 \pm 4\sqrt{5}i}{6} = \frac{-2 \pm 2\sqrt{5}i}{3}\] These solutions are complex and do not exist in the context of real exponential equations.

Interpret the result

Since the quadratic equation has complex roots, the original equation has no real solutions.

Key concepts

Understanding Exponential Equations

Exponential equations are equations where variables appear as exponents. For example, in the equation \(3 \cdot 4^{x}+4 \cdot 2^{x}+8=0\), the variable \(x\) is an exponent. To solve these equations, it's often helpful to express all exponential terms with the same base. This allows for easier manipulation and simplification of the equation.

For instance, in the given exercise, we convert \(4^{x}\) to \(2^{2x}\) and \(8\) to \(2^{3}\). After making these changes, it becomes easier to substitute a single variable for the exponential terms, simplifying the solving process.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations, which have the general form \(ax^2 + bx + c = 0\). The formula is helpful for finding the roots of the equation:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

In this formula, \(a\), \(b\), and \(c\) are coefficients from the quadratic equation. Using this formula, we can find whether the roots are real or complex, depending on the value inside the square root (the discriminant).

In the context of the given exercise, we used a substitution to transform the exponential equation into a quadratic equation: \(3y^2 + 4y + 8 = 0\). By applying the quadratic formula to this new equation, we discovered that the discriminant is negative, leading to complex roots.

Complex Roots

Complex roots occur when the discriminant (inside the square root of the quadratic formula) is negative. In such cases, the solutions involve imaginary numbers. Imaginary numbers are numbers that aren't real and are represented using the symbol \(i\), where \(i = \sqrt{-1}\).

For example, consider the result from our quadratic equation: \[ y = \frac{-4 \pm \sqrt{-80}}{6} \], The negative discriminant (\(-80\)) resulted in two complex roots: \(\frac{-2 \pm 2\sqrt{5}i}{3}\).

These complex roots indicate that the original exponential equation \(3 \cdot 4^{x}+4 \cdot 2^{x}+8=0\) has no real solutions. Real solutions are values of \(x\) that satisfy the original equation within the realm of real numbers, but since we obtained complex roots, it confirms there are no such real values of \(x\) for this equation.