# Precalculus : Write a Quadratic Equation When Given Its Solutions

## Example Questions

### Example Question #1 : Write A Quadratic Equation When Given Its Solutions

Which of the following could be the equation for a function whose roots are at  and ?

Explanation:

If the roots of the equation are at x= -4 and x=3, then we can work backwards to see what equation those roots were derived from. If we factored a quadratic equation and obtained the given solutions, it would mean the factored form looked something like:

Because this is the form that would yield the solutions x= -4 and x=3. If we work backwards and multiply the factors back together, we get the following quadratic equation:

### Example Question #2 : Solving Quadratic Equations

Given roots . Write a quadratic polynomial that has as roots.

Explanation:

We can make a quadratic polynomial with  by mutiplying the linear polynomials they are roots of, and multiplying them out.

Start

Distribute the negative sign

FOIL the two polynomials. This means multiply the firsts, then the outers, followed by the inners and lastly, the last terms.

Simplify

### Example Question #2 : Write A Quadratic Equation When Given Its Solutions

If we know the solutions of a quadratic equation, we can then build that quadratic equation.

Find the quadratic equation when we know that:

and  are solutions.

Explanation:

Since we know the solutions of the equation, we know that:

We simply carry out the multiplication on the left side of the equation to get the quadratic equation.

### Example Question #4 : Solving Quadratic Equations

Which of the following is a quadratic function passing through the points  and ?

Explanation:

These two points tell us that the quadratic function has zeros at , and at .

These correspond to the linear expressions , and .

Expand their product and you arrive at the correct answer.

If the quadratic is opening up the coefficient infront of the squared term will be positive. Thus we get:

.

If the quadratic is opening down it would pass through the same two points but have the equation:

.

Since only  is seen in the answer choices, it is the correct answer.

### Example Question #3 : Write A Quadratic Equation When Given Its Solutions

Which of the following roots will yield the equation .

and

and

Explanation:

When we solve quadratic equations we get solutions called roots or places where that function crosses the x axis. Not all all will cross the x axis, since we have seen that functions can be shifted around, but many will. When they do this is a special and telling circumstance in mathematics. Since we know that roots of these types of equations are of the form x-k, when given a list of roots we can work backwards to find the equation they pertain to and we do this by multiplying the factors (the foil method).

If you were given an answer of the form  then just foil or multiply the two factors. If you were given only two x values of the roots then put them into the form   that would give you those two x values (when set equal to zero) and multiply to see if you get the original function.

For our problem the correct answer is .

Use the foil method to get the original quadratic.

First multiply 2x by all terms in  :

then multiply 2 by all terms in  .

We then combine    for the final answer.

### Example Question #4 : Write A Quadratic Equation When Given Its Solutions

Choose the quadratic equation that has these roots:

and

Explanation:

The roots or solutions of a quadratic equation are its factors set equal to zero and then solved for x. When roots are given and the quadratic equation is sought, write the roots with the correct sign to give you that root when it is set equal to zero and solved. For example, a quadratic equation has a root of -5 and +3. How could you get that same root if it was set equal to zero? With  and  because they solve to give -5 and +3. Thus, these factors, when multiplied together, will give you the correct quadratic equation.

Our roots were .

So our factors are  and .

Now FOIL these two factors:

First:

Outer:

Inner:

Last:

Simplify:

### Example Question #7 : Solving Quadratic Equations

Write the quadratic equation given its solutions.

None of these answers are correct.