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Factoring quadratics is a method used to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\). Factoring involves rewriting the quadratic expression as a product of two binomial expressions. To factor a quadratic equation like \(3x^2 - x - 4 = 0\), it is helpful to first rearrange it into standard form. In this standard form, 'a', 'b', and 'c' are constants corresponding to the coefficients of \(x^2, x\), and the constant term respectively.

After rearranging the equation, the task is to find two numbers that

• multiply to the product of 'a' and 'c', which in this case is \(3 \times -4 = -12\)
• and add up to 'b', which is -1.

In our example, these numbers are 3 and -4. By using these numbers, the quadratic can be factored into \((3x + 4)(x - 1) = 0\). This factorization simplifies the process of finding the solutions or roots of the equation.

After factoring a quadratic equation, solving it involves using the Zero Product Property. This property states that if the product of two factors equals zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero.

For the equation \((3x + 4)(x - 1) = 0\), we break it down into two simpler equations:

• \(3x + 4 = 0\)
• \(x - 1 = 0\)

By solving \(3x + 4 = 0\), we subtract 4 from both sides to obtain \(3x = -4\), then divide by 3 to find \(x = -\frac{4}{3}\). Similarly, solving \(x - 1 = 0\), we add 1 to both sides to get \(x = 1\). These solutions, \(x = -\frac{4}{3}\) and \(x = 1\), are the roots of the quadratic equation.

Root finding for quadratic equations refers to identifying the values of 'x' that make the equation equal to zero. These values are also known as solutions. Once a quadratic equation is successfully factored, finding the roots becomes straightforward. As seen with the quadratic \(3x^2 - x - 4 = 0\), once factored into \((3x + 4)(x - 1) = 0\), each factor can individually be set to zero to find the roots.

• Setting \(3x + 4 = 0\) yields the root \(x = -\frac{4}{3}\).
• Setting \(x - 1 = 0\) yields the root \(x = 1\).

Finding roots is a crucial step in analyzing parabolas' intersections with the x-axis in a graph. The solutions \(x = -\frac{4}{3}\) and \(x = 1\) correspond to the 'zeroes' of the quadratic function, indicating where the parabola crosses or touches the x-axis.