# Algebra II : Finding Roots

## Example Questions

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### Example Question #61 : Solving Quadratic Equations

Find the roots of the quadratic equation

Explanation:

This quadratic equation can be solved in several ways, including the quadratic equation or by graphing. It can also be solved by factoring:

For this quadratic, we are looking for 2 numbers that add to -19 and multiply to 6x10=60. The numbers satisfying these conditions are -4 and -15:

continue factoring

The factors are and . Set each equal to zero:

### Example Question #62 : Solving Quadratic Equations

Find the roots of the quadratic equation

Explanation:

This quadratic can be solved in several different ways, including by graphing or factoring. You can also use the quadratic formula:

### Example Question #63 : Solving Quadratic Equations

Given the equation , find a possible root.

Explanation:

Use the quadratic equation to determine roots of a parabolic function.

The equation  in the form of .

Substitute the known coefficients.

Simplify the equation.

The fraction can be broken into two.

Either  or  is a possible root.

### Example Question #64 : Solving Quadratic Equations

Find the roots of this quadratic:

None of these

Explanation:

Simplify the equation by dividing by 2:

Find the roots using the quadratic equation:

*Because the discriminant (part under the radical) comes to 0, we know our quadratic will have just one repeated solution.

### Example Question #65 : Solving Quadratic Equations

Complex solutions

None of these

Explanation:

Multiply both sides by x:

Put in standard form:

### Example Question #66 : Solving Quadratic Equations

Find the roots:

Explanation:

Evaluate by first writing the quadratic equation.

The given equation is already in standard form.  Write the coefficients.

Simplify the radical using perfect squares.

### Example Question #67 : Solving Quadratic Equations

Find the roots, if any:

Explanation:

Write the quadratic equation in order to solve for the roots.

The coefficients of the parabola  will be used to substitute as values into the given formula.

Substitute the values into the equation.

Simplify the fraction.

Rewrite  using factors of perfect squares.

Replace the term.

### Example Question #68 : Solving Quadratic Equations

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.

To find the zeros, we first need to factor the function. The steps of this process can be seen below.

So, we can see that we have two factors -  and . Now, as usual when solving for zeros, we need to set the right side of our function equal to 0 and solve for . Because if either of these factors is zero, the entire function will equal zero (as anything multiplied by zero is zero), we can consider each factor separately when set to zero.

Thus, we can see that our two zeros are . Since we know that  is when the ball was first launched, we know that when the ball reaches the ground again must be . We can check this by plugging  back into the original function and ensuring that it evaluates to zero.

### Example Question #69 : Solving Quadratic Equations

Evaluate the roots for:

Explanation:

The coefficients of the variables can be determined by the given equation in standard form:  .

Substitute the terms into the equation.

Simplify the terms.

### Example Question #61 : Finding Roots

What is a possible root to the quadratic equation?

Explanation:

The given polynomial  is already in standard form.

Substitute the values into the quadratic formula.

Simplify the top and bottom.

The roots will be complex since we have a negative term inside the radical.

A possible root is:

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