Problem 28 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The solutions are \(x = 2\) and \(x = -1\).

Step 1: Factor out the common term

We start by noticing that the term \(10^x\) can be factored out from each term on the left side of the equation:\[x^2 \cdot 10^x - x \cdot 10^x = 2 \cdot 10^x\]This simplifies to:\[10^x (x^2 - x) = 2 \cdot 10^x\]

Step 2: Cancel common term on both sides

To simplify the equation further, we can divide both sides by \(10^x\), assuming \(10^x eq 0\) which is valid since \(x\) is a real number:\[x^2 - x = 2\]

Step 3: Rearrange the equation

Rearrange the equation into a standard quadratic form by moving all terms to one side of the equation:\[x^2 - x - 2 = 0\]

Step 4: Solve the quadratic equation

Now, solve the quadratic equation \(x^2 - x - 2 = 0\). We use the quadratic formula:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \(a = 1\), \(b = -1\), and \(c = -2\). Calculate the discriminant:\[b^2 - 4ac = (-1)^2 - 4 \times 1 \times (-2) = 1 + 8 = 9\]Find \(x\):\[x = \frac{-(-1) \pm \sqrt{9}}{2 \times 1} = \frac{1 \pm 3}{2}\]This gives two solutions: \(x = 2\) and \(x = -1\).

Step 5: Check the solutions

To ensure the solutions are correct, substitute \(x = 2\) and \(x = -1\) back into the original equation:- For \(x = 2\): \[ 2^2 \cdot 10^2 - 2 \cdot 10^2 = 2 \cdot 10^2 \ 4 \cdot 100 - 2 \cdot 100 = 200 \ 200 - 200 = 200 \] The equation holds.- For \(x = -1\): \[ (-1)^2 \cdot 10^{-1} - (-1) \cdot 10^{-1} = 2 \cdot 10^{-1} \ 1 \cdot 0.1 + 0.1 = 0.2 \] The equation holds.Both solutions are valid.

Key Concepts

Factoring

Factoring is a method used to simplify algebraic expressions and equations by breaking them down into simpler components that multiply together to give the original expression. In the equation given, \[x^2 \cdot 10^x - x \cdot 10^x = 2 \cdot 10^x\],the common factor among the terms on the left-hand side is \(10^x\). By factoring out \(10^x\), we rewrite the equation as:\[10^x(x^2 - x) = 2 \cdot 10^x\].This step reduces the complexity of the equation and makes it simpler to solve.

When you factor, look for:

• Common numbers or variables across terms.
• Patterns like differences of squares or perfect squares.

Factoring helps reveal the internal structure of the equation, making it easier to work with.

Quadratic Formula

The quadratic formula is a universal tool used to solve quadratic equations of the form \[ax^2 + bx + c = 0\].The formula is:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

To use the formula, follow these steps:

• Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
• Substitute these values into the formula.
• Calculate the discriminant, \(b^2 - 4ac\), and find its square root.
• Apply the '+/-' part to find the two possible solutions for \(x\).

This formula is powerful because it works even for complex solutions.In our example, with \(a = 1\), \(b = -1\), and \(c = -2\), substituting these values into the quadratic formula gives us\(x = \frac{1 \pm 3}{2}\),resulting in solutions \(x = 2\) and \(x = -1\).

Discriminant

The discriminant is a key part of the quadratic formula, represented by \(b^2 - 4ac\). It tells us the nature of the roots of a quadratic equation.

Depending on its value:

• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root, also known as a repeated or double root.
• If negative, the solutions are complex numbers.

In our problem, the discriminant is calculated as \[(-1)^2 - 4 \times 1 \times (-2) = 9\].A positive discriminant (9 in our case) indicates two distinct real solutions, confirming that \(x = 2\) and \(x = -1\) are indeed real solutions.

Solving Equations

Solving equations, especially quadratic ones, involves finding values of the variable that make the equation true. The process often involves multiple steps: identifying the type of equation, simplifying it, and applying methods like factoring or quadratic formula as appropriate.

For the equation\[x^2 - x = 2\],we rearranged it into a standard quadratic form\[x^2 - x - 2 = 0\].Once in this form, you choose the best solving strategy: factoring, completing the square, or the quadratic formula.

Checking solutions is crucial. Substituting your results back into the original equation ensures correctness. Both solutions \(x = 2\) and \(x = -1\) checked out when re-inserted into the equation, confirming their validity. Each step in solving ensures thorough understanding and accuracy, making the process an essential part of learning mathematics.