Question

Writing Exercises

Answer 1

Answer:

A quadratic equation can be solved using the quadratic formula, the factoring method, or completing the square. You expect to get either one or two answers.

Step-by-step explanation:

First, we have to understand how to solve a quadratic equation. There are three methods, the first of which is the quadratic formula. For any quadratic equation that takes the standard form of

$f(x)=ax^2+bx+c$

you can solve for x by using the following formula:

x=(-b±√(b^2-4ac))/2a

Sometimes, however, the quadratic expression is factorable, meaning it can be converted into a product of two smaller expressions. For example:

$x^2-4=(x-2)(x+2)$

As you can see, factoring gives you the roots easily (set each smaller expression to 0 and solve for x). If the quadratic is factorable, it would be easiest to use this method first.

Completing the square is another method that can be used to solve a quadratic function. It is often preferred because it converts the function into something factorable. In order to complete the square, you have to first ensure that the quadratic term's coefficient is 1. After that, you can take the linear term's coefficient, divide it by 2, and square it. Take the new term you found and add it to the equation. Because you are adding something to an equation, you must also subtract. Now, you can complete the square and factor. An example can be found below:

$x^{2}-6=0\\\\(x^2-6+9)-9=0\\\\(x-3)^2-9=0\\\\(x-3-3)(x-3+3)=0\\\\x(x-6)=0\\\\x=0, x=6$

A quadratic equation can only have up to two solutions. You can imagine the graph of a quadratic. It looks like a parabola and only changes directions once. This means that it only crosses the x-axis two times. You can also look at the examples given and see that factoring only gives you two smaller expressions (leading to 2 roots). Sometimes, however, you might get a double root, leading to an equation with only one solution. This happens when the discriminant of the quadratic equation is 0. You can calculate this in advance to save yourself some time.