91Ó°ÊÓ

In mathematics, particularly in algebra, we often deal with quadratic equations. A quadratic equation takes the form:
\(ax^2 + bx + c = 0\).
Where \(a\), \(b\), and \(c\) are constants. Factoring quadratics is one method for solving these equations. The basic idea is to rewrite the quadratic as a product of two binomials.
This is only possible if the quadratic can be factored nicely, which is not always the case. Here’s a general approach:

• Look for two numbers that multiply to \(ac\) and add up to \(b\).
• Rewrite the middle term (\(bx\)) using these two numbers.
• Factor by grouping.

For example, take a simpler quadratic like \(x^2 + 5x + 6 = 0\).

• We need two numbers that multiply to \(6\) (product of \(a\) and \(c\)) and add to \(5\).
• These numbers are \(2\) and \(3\).
• Rewrite \(5x\) as \(2x + 3x\): \(x^2 + 2x + 3x + 6 = 0\).
• Factor by grouping: \(x(x + 2) + 3(x + 2) = 0\), which simplifies to \((x + 2)(x + 3) = 0\).

Therefore, the solutions are \(x = -2\) and \(x = -3\).

Completing the square is a useful technique for solving quadratic equations. It transforms a quadratic equation into a perfect square trinomial, which is easier to solve. Here are the steps:

• Start with the quadratic equation in the form \(ax^2 + bx + c = 0\).
• If \(a\) is not 1, divide the whole equation by \(a\) to make the coefficient of \(x^2\) equal to 1.
• Move the constant term \(c\) to the other side of the equation.
• Add the square of half the coefficient of \(x\) to both sides of the equation to complete the square. This transforms the left side into a binomial square.

For the example \(x^2 + 8x - 4 = 0\):

• Rewrite as \(x^2 + 8x = 4\).
• Half of \(8\) is \(4\), and its square is \(16\). Therefore, add 16 to both sides:
  \(x^2 + 8x + 16 = 4 + 16\).
• This simplifies to \((x + 4)^2 = 20\).

Next, take the square root of both sides and solve for \(x\): \(x + 4 = \pm \sqrt{20}\), which gives \(x = -4 \pm 2\sqrt{5}\).

A perfect square trinomial is a special quadratic expression that can be written as the square of a binomial. It has the form \((x + p)^2 = x^2 + 2px + p^2\). Recognizing and creating perfect square trinomials is crucial for solving quadratics by completing the square.
For the quadratic \(x^2 + 8x + 16 = 20\), we rewrote it to \((x + 4)^2 = 20\). This shows that \(x^2 + 8x + 16\) is indeed a perfect square trinomial: \(x^2 + 8x + 16\).
This factorization helps us solve the equation easily.
The steps involved are:

• Identify the quadratic expression.
• Rewrite it as \((x + p)^2\), where \(p\) is half the coefficient of \(x\).
• Simplify and solve for \(x\).

To solidify this concept, let's take another example: \(x^2 + 6x + 9\).
Half of \(6\) is \(3\) and \(3^2\) is \(9\), making this a perfect square trinomial: \((x + 3)^2\). So the equation is transformed and can be easily solved if set equal to another value.