# Algebra II : Quadratic Formula

## Example Questions

### Example Question #41 : Quadratic Formula

Find a root for:

Explanation:

Write the quadratic equation that applies for .

There is no  term.   Substitute the known coefficients from the polynomial.

Simplify the numerator and denominator.

The roots will be imaginary.

One of the possible answers is:

### Example Question #42 : Quadratic Formula

Solve:

None of these

Explanation:

Since our quadratic is in standard form , just plug in the values from the equation.

Divide both numerators by the denominator to simplify:

### Example Question #43 : Quadratic Formula

Use the quadratic formula to determine a root:

Explanation:

Write the quadratic formula for polynomials in the form of

Substitute the known values.

Simplify the equation.

The roots to this parabola is:

### Example Question #44 : Quadratic Formula

Find a root using the quadratic equation:

Explanation:

Rewrite the given equation in  form.

We can determine the coefficients of the terms.

Substitute these values into the quadratic equation.

Rewrite this fraction using common factors, and simplify each step.

One of the possible answers given is:

### Example Question #45 : Quadratic Formula

Solve the following equation:

Explanation:

Let . Then the given equation can be rewritten in terms of  as follows:

.

Since , we now have  and , which implies that  and . Substituting each of these values yields a true statement; hence, the solutions to the original equation are

.

### Example Question #46 : Quadratic Formula

Solve for the roots:

Explanation:

The given equation  is in standard form of:

The coefficients correspond to the values that go inside the quadratic equation.

Substitute the values into the equation.

Simplify this equation.

The radical can be rewritten as:

Substitute and simplify the fraction.

### Example Question #41 : Quadratic Formula

Solve for the roots, if any:

Explanation:

This polynomial is in the standard form of a parabola, .

Identify the variables.

Split the fraction.

### Example Question #48 : Quadratic Formula

Determine the roots, if possible:

Explanation:

This equation will apply for polynomials in  format.

Substitute the correct values into the quadratic formula.

Simplify this equation.

### Example Question #49 : Quadratic Formula

A ball is launched straight up from the ground at time , and the height of the ball from the ground at time  is described by the function . Find the time at which the ball returns to the ground.

Explanation:

In order to solve this problem, we think about how to translate what is asked of us in the problem into a mathematical notion. The problem asks us to find when the ball returns to the ground. We know that the ball is launched from the ground at time , and if we plug  into our original function, we will find that at time ,  . Therefore, our "ground" is located at , which is also the horizontal axis. This means that if we are looking for when our ball returns to the ground, mathematically we are looking for values of  which make . In other words, we are looking for the zeros (or horizontal intercepts) of our function.

Typically, the easiest way to find zeros is to factor, but in this situation factoring won't get us very far. So, we have to turn the quadratic formula. Remember for a function f(x) in the standard quadratic form, , you can find the zeros by plugging them into the quadratic formula, which is...

The first step in using the quadratic formula is to identify the proper , and . In our case,  and , while . So, to solve for our zeros, we simply need to plug these numbers into the quadratic formula and solve for x.

Since we know that  is when our ball was launched, we know that  must be when our ball returns to the ground.

### Example Question #50 : Quadratic Formula

Find the zeros of the function f(x) where