Problem 9 - Algebra 2 and Trigonometry

Question

Solve and check each of the equations. \(4-x(x-3)=0\)

Step-by-Step Solution

Answer

The solutions are \(x = 4\) and \(x = -1\).

Step 1: Distribute the x

First, we need to eliminate the parentheses by distributing the variable inside of them. The equation is: \[ 4 - x(x - 3) = 0 \] Distribute the \(x\) to each term inside the parenthesis: \[ x(x - 3) = x^2 - 3x \] Then, substitute back into the equation: \[ 4 - (x^2 - 3x) = 0 \]

Step 2: Simplify the expression

Now, simplify the expression. Distribute the negative sign across the parenthesis: \[ 4 - x^2 + 3x = 0 \] Rearrange the terms to match the standard form of a quadratic equation: \[ -x^2 + 3x + 4 = 0 \] To make the leading coefficient positive, multiply the entire equation by -1: \[ x^2 - 3x - 4 = 0 \]

Step 3: Factor the quadratic equation

Now that we have a standard quadratic equation, we need to find solutions by factoring. Look for two numbers that multiply to \(-4\) and add to \(-3\): These numbers are \(-4\) and \(+1\). So, we can factor the quadratic as: \[ (x - 4)(x + 1) = 0 \]

Step 4: Solve for x

Now, set each factor equal to zero and solve for \(x\) to find the solutions: 1. \(x - 4 = 0\): \[ x = 4 \] 2. \(x + 1 = 0\): \[ x = -1 \] Thus, the solutions are \(x = 4\) and \(x = -1\).

Step 5: Check the solutions

To ensure correctness, substitute each solution back into the original equation to check: For \(x = 4\): \[ 4 - 4(4 - 3) = 4 - 4 \cdot 1 = 4 - 4 = 0 \] Correct! For \(x = -1\): \[ 4 - (-1)(-1 - 3) = 4 - (-1)(-4) = 4 - 4 = 0 \] Correct! Since both solutions satisfy the original equation, they are verified.

Key Concepts

Factoring

Factoring is a simple method used to solve quadratic equations. The goal is to express a quadratic equation in the form of \[ ax^2 + bx + c = 0 \] as a product of two binomials, which looks like \[ (x + p)(x + q) = 0. \]This process involves finding two numbers, \(p\) and \(q\), that multiply to the constant term \(c\) and add to the linear coefficient \(b\). In this example, the quadratic equation \[ x^2 - 3x - 4 = 0 \] was factored as \[ (x - 4)(x + 1) = 0. \]Both \(-4\) and \(+1\) multiply to \(-4\) and add up to \(-3\), successfully completing the factorization.Factoring transforms the equation into easier linear ones:

• Each binomial is set to zero: \(x - 4 = 0\) and \(x + 1 = 0\).
• It provides the simple solutions for \(x\): \(x = 4\) or \(x = -1\).

This is why factoring is such a useful and quick method for solving quadratic equations.

Quadratic Equation

A quadratic equation is any equation of the form: \[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, and \(x\) represents an unknown.Quadratics are central in algebra because they model various real-world phenomena such as projectile motion or area problems.
A key feature of quadratics is their parabolic graph that opens upwards or downwards depending on the leading coefficient \(a\). More complex equations will start off non-factored, like our original: \[ 4 - x(x - 3) = 0. \]Transforming this into the standard quadratic form requires a few steps:

• Distributing \(x\) across terms: \(x^2 - 3x\).
• Flattening terms and making \(x^2\) positive gives \[ x^2 - 3x - 4 = 0. \]

This standard form is essential as it allows easy application of the factoring method, and other methods like completing the square or using the quadratic formula.

Checking Solutions

Checking solutions ensures the results are correct, a vital step after solving quadratic equations.
Substituting the solutions back into the original equation verifies their accuracy. For example, after solving \[ x^2 - 3x - 4 = 0 \]we found \(x = 4\) and \(x = -1\). By substituting these values back into the original equation \[ 4 - x(x - 3) = 0 \],we confirm:

• For \(x = 4\): \[ 4 - 4(4 - 3) = 0. \] This simplifies to \[ 4 - 4 = 0. \]
• For \(x = -1\): \[ 4 - (-1)(-1 - 3) = 0. \] Simplifying gives \[ 4 - 4 = 0. \]

Both solutions satisfy the original equation, proving their correctness. Checking solutions not only ensures accuracy but also helps reinforce understanding.