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The quadratic formula is a universal tool used to find the roots of any quadratic equation. A quadratic equation is of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The formula itself is given by: \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This equation provides two solutions for \( x \), which can be real or complex.

• \( -b \) is the opposite of \( b \), which shifts the parabola horizontally.
• \( \sqrt{b^2 - 4ac} \) is called the discriminant. It determines the nature of the roots.
• If the discriminant is positive, the equation has two distinct real roots.
• If it is zero, there is exactly one real root.
• If it's negative, the roots are complex, involving imaginary numbers.

In our solution, the discriminant was negative (\( -87 \)), which indicates complex roots. This is why the solution included \( 3i\sqrt{5} \), with \( i \) being the imaginary unit.

The FOIL method is an acronym that stands for First, Outer, Inner, Last. It's a technique used to multiply two binomials. Let's apply it to the equation from our problem: - \((2x-6)(x+2)\) Using FOIL, we multiply the terms:

• First: Multiply the first terms: \( 2x \cdot x = 2x^2 \).
• Outer: Multiply the outer terms: \( 2x \cdot 2 = 4x \).
• Inner: Multiply the inner terms: \( -6 \cdot x = -6x \).
• Last: Multiply the last terms: \( -6 \cdot 2 = -12 \).

Add all these results together to get \( 2x^2 + 4x - 6x - 12 \), simplifying to \( 2x^2 - 2x - 12 \). The FOIL method is straightforward and helps simplify expressions before solving quadratics.

Completing the square is another method used to solve quadratic equations. The idea is to convert the equation into a perfect square trinomial, making it easier to solve. Here’s a simple explanation of completing the square:

• First, ensure the coefficient of \( x^2 \) is 1. If not, divide through by this coefficient.
• Move the constant term to the right side of the equation.
• Add \( \left( \frac{b}{2} \right)^2 \), where \( b \) is the coefficient of \( x \), to both sides.
• This transforms the left-hand side into a perfect square trinomial \( \left(x + \frac{b}{2}\right)^2 \).

This technique was not used in solving \( 2x^2 - 7x + 29 = 0 \) but is helpful when the quadratic formula seems complicated or if you need to graph the equation by plotting its vertex.

Complex numbers arise when solving equations that have no real solutions. They are written in the form \( a + bi \), where \( i \) is the imaginary unit and \( i^2 = -1 \). While real numbers lie on the number line, complex numbers can be visualized as points in a two-dimensional plane.- In the quadratic formula, complex roots occur when the discriminant \( b^2 - 4ac \) is negative. For example, in our equation, the discriminant \( 49 - 232 = -87 \) was negative, which resulted in complex solutions. Thus, the solutions for \( x \) were \( \frac{7 \pm 3i\sqrt{5}}{4} \). In this result:

• The real part is \( \frac{7}{4} \).
• The imaginary part is \( \pm \frac{3i\sqrt{5}}{4} \), where the \( i \) denotes the imaginary component.

Understanding complex numbers is crucial as it extends the concept of numbers beyond the real line, allowing us to solve a broader range of equations.