Questions: Solve the quadratic by factoring: x^2 + 14x + 4 = 4(x - 3)

Solve the quadratic by factoring: x^2 + 14x + 4 = 4(x - 3)
Transcript text: Solve the quadratic by factoring: \[ x^{2}+14 x+4=4(x-3) \]
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Solution

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Solution Steps

To solve the quadratic equation by factoring, we first need to simplify and rearrange the equation to standard form \(ax^2 + bx + c = 0\). Then, we can factor the quadratic expression and solve for the values of \(x\).

Solution Approach
  1. Expand and simplify the given equation.
  2. Rearrange the equation to standard form \(ax^2 + bx + c = 0\).
  3. Factor the quadratic expression.
  4. Solve for the values of \(x\).
Step 1: Expand and Simplify the Given Equation

Given the equation: \[ x^2 + 14x + 4 = 4(x - 3) \] First, expand the right-hand side: \[ 4(x - 3) = 4x - 12 \] So the equation becomes: \[ x^2 + 14x + 4 = 4x - 12 \]

Step 2: Rearrange to Standard Form

Rearrange the equation to the standard form \(ax^2 + bx + c = 0\): \[ x^2 + 14x + 4 - 4x + 12 = 0 \] Combine like terms: \[ x^2 + 10x + 16 = 0 \]

Step 3: Factor the Quadratic Expression

Factor the quadratic expression \(x^2 + 10x + 16\): \[ x^2 + 10x + 16 = (x + 8)(x + 2) \]

Step 4: Solve for \(x\)

Set each factor equal to zero and solve for \(x\): \[ x + 8 = 0 \quad \Rightarrow \quad x = -8 \] \[ x + 2 = 0 \quad \Rightarrow \quad x = -2 \]

Final Answer

\(\boxed{x = -8, -2}\)

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