To find the roots, or solutions, of the equation, factor the quadratic. It factors to . Then, set each expression equal to 0 to get your roots of 1 and 4.

An x-intercept is the same thing as a root or solution. Therefore, we can set the function equal to 0 and solve for x. .

The first step in finding the root is to factor the polynomial. The common factor in each term is , so extract that from the equation first. . Continue to find the numbers that will multiply to , and add to . If there is no number in front of the variable, it means there is just one of them! 

The factors of . One of the terms will need to be negative, and in this case it will be the larger number in order to equal . The numbers that work are . The factored polynomial would be . There are three roots for this problem. The first will be 0, because if the  is multiplied by , then it will equal 0. The other numbers to fulfill the equation would be . Therefore, the roots are .

Roots are the values of "x" where your function equals 0. You need to set both equations equal to 0 and then solve for "x." Upon doing so you get:

First, set the equation equal to zero.

From here, factor the equation into two binomials.

Now, set each binomial equal to zero and solve for x.

 AND 

 AND 

 AND   

Pull out an 

Then factor

So,

To find the roots of this function, we must set it equal to zero and solve for x:

Now, we must find two factors of 12 that add together to get -7. These numbers are -4 and -3. 

We can rewrite this function, then, as two binomials multiplied together:

Now, because this product equals zero, each of the binomials can be set equal to zero:

Our two roots are .

This problem could be worked out using the quadratic formula, but in this particular case it's easier to factor the left side. 

 and  are the roots that zero the expression on the left side of the equation. In the graph, the curve - which happens to be a paraboloa - will cross the x-axis at the roots. 

A few more points...

Observe that the coefficient for the  term in the original quadratic is the sum of   and . Also, the constant term in the originl equation is the product of    and . It's a good rule of thumb to look for numbers that will satsify these conditions when you are setting off to solve a quadratic. Observe how this happens, 

If you notice this pattern in a quadratic, then factoring is always a faster approach. The quadratic formula will always work too, but may take a little longer.

Unfortunately you will often find that factoring is not an option since you will not always be abe to easily find such a pattern for most quadratics, especially if the roots are not whole number integers, or if one or both of the roots are complex numbers. 

Step 1: Rewrite denominator of 2nd term in factored form

Step 2: Multiply each term by the common denominator

Step 3: Simplify

Step 4: Combine like terms, set equal to zero

Step 5: Factor & solve

To find the roots of a quadratic function, we must find the x values where the function is equal to zero. To do this, we must set the function equal to zero:

Now, we factor:

The factoring can be done using a number of methods.

Now, set each binomial equal to zero and solve for x: