Problem 87 - A Graphical Approach to College Algebra

Question

$$\text { The given equations are quadratic in form. Solve each and give exact solutions.}$$ $$3^{2 x}+35=12\left(3^{x}\right)$$

Step-by-Step Solution

Answer

Solutions: \(x = \log_3{7}\) and \(x = \log_3{5}\).

Step 1: Recognize the Substitution Form

Identify that the equation \(3^{2x} + 35 = 12(3^x)\) is quadratic in form. We can write \(3^{2x}\) as \((3^x)^2\). Let \(y = 3^x\). This simplifies our equation to \(y^2 + 35 = 12y\).

Step 2: Rearrange to Standard Quadratic Form

Rearrange the equation \(y^2 + 35 = 12y\) to the standard quadratic form. Subtract \(12y\) from both sides to obtain \(y^2 - 12y + 35 = 0\).

Step 3: Solve the Quadratic Equation

Solve the quadratic equation \(y^2 - 12y + 35 = 0\) using the quadratic formula: \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). Here, \(a = 1\), \(b = -12\), and \(c = 35\).

Step 4: Calculate the Discriminant

Calculate the discriminant \(b^2 - 4ac\). Substitute the values: \((-12)^2 - 4(1)(35) = 144 - 140 = 4\). The discriminant is 4.

Step 5: Find the Solutions for y

Using the quadratic formula with the discriminant value 4, find the solutions: \( y = \frac{12 \pm \sqrt{4}}{2} \). This simplifies to \( y = \frac{12 \pm 2}{2} \). Thus, \( y = 7 \) or \( y = 5 \).

Step 6: Back-Substitute to Find x

Recall that \(y = 3^x\). For \(y = 7\), solve \(3^x = 7\) to find \(x = \log_3{7}\). For \(y = 5\), solve \(3^x = 5\) to find \(x = \log_3{5}\).

Step 7: Express the Solutions

The exact solutions for the original equation are \(x = \log_3{7}\) and \(x = \log_3{5}\). These are the values that satisfy the equation when substituting back into the exponential form.

Key Concepts

Exponential Equations

Exponential equations are mathematical expressions where variables appear as the exponents. They are commonly of the form \( a^x = b \), where \( a \) is the base and \( b \) is a constant. Solving exponential equations often involves adjusting bases or using logarithms.
In the provided exercise, the equation \( 3^{2x} + 35 = 12(3^x) \) is an exponential equation because the variable \( x \) is in the exponent.
To solve, we first recognize that the equation can take the form \( (3^x)^2 \), simplifying to a quadratic expression with respect to \( 3^x \).

• An initial step might involve manipulating the equation to get a common base on one side to simplify the expression.
• If necessary, logarithms can be used for a more straightforward solution when finding exact values of \( x \).

Recognizing exponential elements helps in transforming the equation into a solvable form using simpler algebraic techniques.

Substitution Method

The substitution method is a powerful technique for simplifying and solving complex equations, particularly when dealing with exponents. It involves replacing a variable expression with a simpler variable.
In the original problem, we used substitution by letting \( y = 3^x \). This transforms the equation from an exponential form \( 3^{2x} + 35 = 12(3^x) \) to a quadratic form \( y^2 + 35 = 12y \).

• The primary purpose of substitution is to reduce the equation to a more manageable form, making it easier to apply familiar solving techniques like the quadratic formula.
• Substitution is especially useful when equations appear to be in quadratic form upon inspection but have exponential expressions.

Substituting back at the end finds the original variable's value, connecting the simpler solution to the original equation.

Discriminant

The discriminant in a quadratic equation \( ax^2 + bx + c = 0 \) is the expression \( b^2 - 4ac \). It provides vital information about the nature of the roots of the equation.
For the equation \( y^2 - 12y + 35 = 0 \), the discriminant is calculated as \( (-12)^2 - 4(1)(35) = 144 - 140 = 4 \).

• A positive discriminant indicates two distinct real roots.
• A discriminant of zero indicates a single repeated real root.
• A negative discriminant indicates no real roots but two complex roots.

In this case, the discriminant is positive (4), meaning the quadratic has two distinct real roots. Knowing the discriminant before solving assists in predicting the type of solutions obtained. It simplifies anticipating whether additional solving techniques, like factoring, can be employed.

Quadratic Formula

The quadratic formula is a universal method for solving quadratic equations of the form \( ax^2 + bx + c = 0 \). The formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
In this exercise, after using the substitution \( y = 3^x \), we obtain a quadratic \( y^2 - 12y + 35 = 0 \).

• Assigning the coefficients as \( a = 1 \), \( b = -12 \), \( c = 35 \), we substitute them into the quadratic formula. This gives: \( y = \frac{12 \pm \sqrt{4}}{2} \).
• Simplifying the expression, we obtain the solutions for \( y \): \( y = 7 \) and \( y = 5 \).
• These solutions indicate the values that satisfy the quadratic equation.

Substitution of these values back into the exponential form \( 3^x = y \) solves for \( x \) in terms of logarithms: \( x = \log_3(7) \) and \( x = \log_3(5) \). The quadratic formula is efficient for obtaining roots when other methods are not easily applicable.