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The Greatest Common Factor, or GCF, is an essential concept when it comes to factoring. It's the largest number that divides each of the terms in an expression without leaving a remainder. In solving quadratic equations, identifying and factoring out the GCF can simplify the expression, making the equation easier to handle. In the equation \(48x^2 + 12x - 90 = 0\), we determine that the coefficients are 48, 12, and -90. The GCF of these numbers is 6, meaning 6 is the largest number that can be divided evenly into each coefficient. By factoring out the GCF, the equation becomes much simpler:

• Original equation: \(48x^2 + 12x - 90 = 0\)
• Factored form: \(6(8x^2 + 2x - 15) = 0\)

Remember, factoring out the GCF is just the first step. It's like clearing away unnecessary clutter so that you can focus on factoring the more manageable trinomial within the parentheses.

Factoring trinomials is a special technique used to break down quadratic expressions into simpler factors that can be easily solved. A trinomial is typically in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.In the example \(8x^2 + 2x - 15\), we need to find two numbers that multiply to the product of \(a \times c = 8 \times (-15) = -120\) and add up to \(b = 2\). This step involves a bit of trial and error, and it might be helpful to write a list of factor pairs for -120.

• Two numbers that satisfy these conditions here are 12 and -10.

By rewriting the trinomial using these numbers, we can move onto the next step in the factoring process.

After factoring the quadratic equation, the next step is to solve it by finding the values of \(x\) that satisfy the equation. Once you've factored the quadratic expression, you will have it in a form like \((4x - 5)(2x + 3) = 0\).To solve for \(x\), set each factor equal to zero. This is because if the product of two numbers equals zero, at least one of the numbers must be zero.

• For the factor \(4x - 5\), set it equal to zero: \(4x - 5 = 0 \Rightarrow 4x = 5 \Rightarrow x = \frac{5}{4}\).
• For the factor \(2x + 3\), set it equal to zero: \(2x + 3 = 0 \Rightarrow 2x = -3 \Rightarrow x = -\frac{3}{2}\).

These solutions are the roots of the quadratic equation, meaning the equation holds true when \(x\) is either \(\frac{5}{4}\) or \(-\frac{3}{2}\).

Factoring by grouping is a method used to factor polynomials with four terms. After rewriting the trinomial \(8x^2 + 2x - 15\) as \(8x^2 + 12x - 10x - 15\), you can look at the expression as two separate groups.Group the terms:

• \((8x^2 + 12x) + (-10x - 15)\)

Now, factor out the GCF from each group separately:

• From \(8x^2 + 12x\), you can factor out \(4x\), resulting in \(4x(2x + 3)\).
• From \(-10x - 15\), factor out \(-5\), resulting in \(-5(2x + 3)\).

Notice the common factor \((2x + 3)\) in both groups, allowing you to factor by grouping:

• \((4x - 5)(2x + 3) = 0\)

This technique can simplify complex expressions into more manageable parts, revealing the solutions hidden within.