Problem 34 - Precalculus Mathematics for Calculus

Question

Solve the equation. $$x^{2} 10^{x}-x 10^{x}=2\left(10^{x}\right)$$

Step-by-Step Solution

Answer

Solutions are \(x = 2\) and \(x = -1\).

Step 1: Factor Common Terms

First, observe that both terms on the left-hand side of the equation contain the common factor \(10^x\). Therefore, we can factor \(10^x\) out from the left side of the equation. This gives us:\[ (x^2 - x)10^x = 2 \times 10^x \]

Step 2: Divide Both Sides by \(10^x\)

Since \(10^x\) is a common factor and can be canceled out (as long as \(10^x eq 0\), which is always true for real \(x\)), divide both sides by \(10^x\):\[ x^2 - x = 2 \]

Step 3: Rearrange Terms into a Standard Quadratic Equation

Rearrange the equation to form a standard quadratic equation:\[ x^2 - x - 2 = 0 \]

Step 4: Factor the Quadratic Expression

We need to factor the quadratic equation \(x^2 - x - 2 = 0\). We look for two numbers that multiply to \(-2\) and add up to \(-1\). These numbers are \(-2\) and \(1\). Therefore, we can factor the equation as:\[ (x - 2)(x + 1) = 0 \]

Step 5: Solve for x

Using the zero-product property, set each factor equal to zero and solve for \(x\):1. \(x - 2 = 0 \rightarrow x = 2\)2. \(x + 1 = 0 \rightarrow x = -1\)

Step 6: Verify Solutions

Substitute \(x = 2\) and \(x = -1\) back into the original equation to check if they satisfy both sides:- For \(x = 2\): \[ (2)^2 \cdot 10^2 - 2 \cdot 10^2 = 2 \cdot 10^2 \] \(40 - 20 = 20\), which is true.- For \(x = -1\): \[ (-1)^2 \cdot 10^{-1} - (-1) \cdot 10^{-1} = 2 \cdot 10^{-1} \] \(0.1 + 0.1 = 0.2\), which is true.

Key Concepts

Factoring Quadratics

Factoring quadratics is a crucial skill when solving quadratic equations. It involves breaking down a quadratic expression, such as \( x^2 - x - 2 \), into a product of simpler binomials. This process can often simplify solving the equation. To factor a quadratic expression, you look for two numbers that multiply to the constant term (in this case, -2) and add to the linear coefficient (in this case, -1).

• The two numbers that meet these criteria are -2 and 1.
• They multiply to -2 and add up to -1.

Using these numbers, the expression \( x^2 - x - 2 \) can be factored into \((x - 2)(x + 1)\).
Factoring simplifies the quadratic into a form where the Zero-Product Property can easily be applied to find solutions.

Zero-Product Property

The Zero-Product Property is a fundamental principle in algebra, which states that if a product of factors is zero, then at least one of the factors must be zero. This property is especially useful in solving quadratic equations that have been factored.
For example, in the equation \((x - 2)(x + 1) = 0\), the Zero-Product Property allows us to set each factor equal to zero:

• Set \(x - 2 = 0\), which gives the solution \(x = 2\).
• Set \(x + 1 = 0\), which gives the solution \(x = -1\).

By applying this property, we simplify our task of solving the quadratic equation into solving two simpler linear equations. This approach is quick and efficient, making it a powerful tool in algebra.

Verification of Solutions

Verification of solutions is an essential step in solving equations. It ensures that the solutions obtained satisfy the original equation.
To verify solutions, substitute each value back into the original equation to see if it makes the equation true.

• For \(x = 2\), substituting back into the original equation confirms that both sides balance as \(40 - 20 = 20\).
• For \(x = -1\), make sure both sides match as \(0.1 + 0.1 = 0.2\).

Verification confirms the correctness of the solutions, providing confidence in the results obtained through prior steps like factoring and applying the Zero-Product Property. This process eliminates errors and assures accuracy.