Precalculus : Graphing Functions

Study concepts, example questions & explanations for Precalculus

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Example Questions

Example Question #1 : Solve A Quadratic Equation

Solve the following quadratic equation:

Possible Answers:

Correct answer:

Explanation:

When solving a quadratic equation, the first thing to look for is whether or not it can be factored, as this is most often the easiest and fastest method if the quadratic can in fact be factored. We can see that each of the terms in the given equation have a common factor of 3, so it will be easier to factor the quadratic if we first factor out the 3:

Now we're left with a polynomial where we need to find two numbers whose product is -28 and whose sum is -3. Thinking about the factors of 28, we can see that 4 and 7 will yield -3 if 7 is negative and 4 is positive, so we now have our factorization:

Example Question #2 : Solve A Quadratic Equation

Find the root(s) of the following quadratic equation?

Possible Answers:

Correct answer:

Explanation:

To find the roots of an equation in the form , you use the quadratic formula 

.

In our case, we have

This gives us  which simplifies to 

Example Question #3 : Solve A Quadratic Equation

Given the function , find a possible root for this quadratic.

Possible Answers:

Correct answer:

Explanation:

Factorize  and set this equation equal to zero.

The answer  is one of the possible choices.

Example Question #4 : Solve A Quadratic Equation

Solve the quadratic equation for .

Possible Answers:

None of the other answers.

Correct answer:

None of the other answers.

Explanation:

There are two solutions; .

We proceed as follows. 

Add  to both sides.

 

Take the square root of both sides, remember to introduce plus/minus on the right side since you are introducing a square root into your work.

 

Add  to both sides.

 

Example Question #5 : Solve A Quadratic Equation

Solve the quadratic equation for .

Use the quadratic formula.

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

For any quadtratic equation of the form , the quadratic formula is

Plugging in our given values  we have:

Example Question #6 : Solve A Quadratic Equation

Find the roots of the equation.

Possible Answers:

Correct answer:

Explanation:

Use either the quadratic formula or factoring to solve the quadratic equation.

Using factoring, we want to find which factors of six when multiplied with the factors of two and then added together result in negative one.

Using the quadratic formula,

let 

Example Question #7 : Solve A Quadratic Equation

Solve .

Possible Answers:

Correct answer:

Explanation:

To solve this equation, use trial and error to factor it. Since the leading coefficient is , there is only one way to get  , so that is helpful reminder. Once it's properly factored, you get: . Then, set both of those expressions equal to  to get your roots: .

Example Question #1 : Solve A Polynomial By Factoring

Solve the following polynomial for  by factoring:

Possible Answers:

Correct answer:

Explanation:

The polynomial in the problem is given as follows:

Factoring this polynomial, we would get an expression of the form:

So we need to determine what a and b are. We know we need two factors that when multiplied equal -12, and when added equal -1. If we consider 2 and 6, we could get -12 but could not arrange them in any way that would make their sum equal to -1. We then look at 3 and 4, whose product can be -12 is one of them is negative, and whose sum can be -1 if -4 is added to 3. This tells us that the 4 must be the negative factor and the 3 must be the positive factor, so we get the following:

Example Question #2 : Solve A Polynomial By Factoring

Solve the following polynomial for  by factoring:

Possible Answers:

Correct answer:

Explanation:

Factoring our polynomial, we can see we will have 2x and x at the beginning of each factor, while we need to find two numbers whose product is 15 and whose sum when multiplied by our leading terms and added is -11x. This gives us the following factorization:

So now we can set each term equal to 0 and solve for our two values of x:

Example Question #1 : Solve A Polynomial By Factoring

Find the root(s) for the function:  

Possible Answers:

Correct answer:

Explanation:

The function  is in the form , and can be factorized.

Determine the values of  and .

Substitute this into the formula.

Set  to find all roots.

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