The quadratic formula:

If  is one odd number, then the next odd number is . If their product is 143, then the following equation is true.

Distribute into the parenthesis.

Subtract 143 from both sides.

This can be solved by factoring, or by the quadratic equation. We will use the latter.

We are told that both integers are positive, so .

The other integer is .

To factor this equation, first find two numbers that multiply to 35 and sum to 12.  These numbers are 5 and 7.  Split up 12x using these two coefficients:

To solve this, we look at the equation .

Setting the equation equal to 0 we get .

Once in this form, we can use the Quadratic Formula to solve for .

The quadratic formula says that if , then 

.

Plugging in our values:

Therefore or  and since we are looking only for positive values (because we can't have negative time), 3.4375 seconds is our answer.

First we want to put the equation into standard form; we do this by making sure the equation is = 0, so let's subtract 4 from both sides.

We could go straight to the quadratic formula from here, but quadratics are always easier to solve if we eliminate Greatest Common Factors first. In this case the GCF is 4 so let's divide both sides by 4.

Now we can compare against the standard form to find a, b, and c.

By pattern matching, we see that a = 1, b = -3, and c = 4. Now we substitute into the quadratic formula.

We can evaluate the square root of a negative number by factoring out the square root of -1 and calling it the imaginary number i. This gives our answer.

Since the equation is already in standard form, we can compare it to the general version of standard form to find a, b, and c.

By pattern matching it is clear that a = 2, b = 4, and c = -3. Now we can substitute into the quadratic formula.

Recall that when subtracting a negative number [as in - 4(2)(-3)], the result is addition:

Combine terms and simplify the radical:

Now eliminate common factors to reduce to lowest terms.

We begin by designating values of a, b and c by comparing the equation to the standard form.

By pattern matching it is clear that a = 2, b = -3, and c = -5. We can now substitute into the quadratic formula:

Take note that in the two cases where negatives/minus signs are multiplied together [-(-3) and -4(2)(-5)], they bceome positive:

Now we simplify and evaluate.

Note that in the previous step we listed the subtraction instance first, as that instance yields the smaller number and it is usually convenient to start sets with smaller numbers first. 

Recall that the quadratic formula is defined as:

For this question, the variables are as follows:

Substituting these values into the equation, you get:

Since you have to take the square root of a negative number, this means that there are no real solutions.

Recall that the quadratic formula is defined as:

For this question, the variables are as follows:

Substituting these values into the equation, you get:

Use a calculator to determine the final values.

Recall that the quadratic formula is defined as:

For this question, the variables are as follows:

Substituting these values into the equation, you get:

Use a calculator to determine the final values.