X2 +4x-12

To work with the quadratic expression \( x^2 + 4x - 12 \), you might want to factor it, complete the square, or use the quadratic formula to find its roots. Let's go through these steps.

Factorization

1. Find two numbers that multiply to -12 (the constant term) and add to 4 (the coefficient of \( x \)). The numbers are 6 and -2.

2. Write the expression as a product of two binomials: \[ x^2 + 4x - 12 = (x + 6)(x - 2) \]

Completing the Square

1. Start with the original equation: \[ x^2 + 4x - 12 \]

2. Move the constant to the other side: \[ x^2 + 4x = 12 \]

3. Take half of the coefficient of \( x \) (which is 4), square it, and add it to both sides: \[ \left(\frac{4}{2}\right)^2 = 4 \quad \Rightarrow \quad x^2 + 4x + 4 = 12 + 4 \]

4. Re-write it: \[ (x + 2)^2 = 16 \]

5. Take the square root of both sides: \[ x + 2 = \pm 4 \]

6. Solving for \( x \) gives: \[ x + 2 = 4 \quad \Rightarrow \quad x = 2 \] \[ x + 2 = -4 \quad \Rightarrow \quad x = -6 \]

Quadratic Formula

For a general quadratic equation of the form \( ax^2 + bx + c = 0 \), the roots can be found using the formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 1 \), \( b = 4 \), \( c = -12 \).

1. Calculate the discriminant: \[ b^2 - 4ac = 4^2 - 4 \cdot 1 \cdot (-12) = 16 + 48 = 64 \]

2. Substitute into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{64}}{2 \cdot 1} = \frac{-4 \pm 8}{2} \]

3. Calculate the two solutions: \[ x = \frac{4}{2} = 2 \quad \text{and} \quad x = \frac{-12}{2} = -6 \]

Summary of Solutions

The roots of the quadratic equation \( x^2 + 4x - 12 = 0 \) are:

• \( x = 2 \)
• \( x = -6 \)

The expression can also be factored as \( (x + 6)(x - 2) \).