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May 25th

May 25th

By the end of this section, you should be able to solve the following problems.
1. Use the quadratic formula to solve the equation.

t2 + 5t = 4

2. Use the quadratic formula to solve the equation.

5x2 + 7x − 2 = 0

3. Use the quadratic formula to solve the equation.

2x2 − 5x+2 = 0

4. Solve the application problem.
In a group of children , each child gives a gift to every other child, If
the number of gifts is 132, find the number of children.

## 2 Concepts

A quadratic equation is an equation in one variable where the highest integer
power of the variable is 2. The general quadratic equation written in standard
form is:

ax2 + bx + c = 0

The solution of this equation can be derived by algebraic methods and it
is:

The way we use this equation is to arrange a specific example in standard
form, and then plug the coefficients into the equation above. Below we will

## 3 Example

3x2 − x = 10

First we put the equation in standard form by adding -10 to both sides
to get:

3x2 − x − 10 = 0

The coefficient of the quadratic term is always a. Here, a=3. The
coefficient of the linear term , x, is always b. Here, b=1. The constant
term is always c. Here, c=-10. We now proceed to solve this equation
by substituting the coefficients into the equation.

Therefore, x = 2 or x = −5/3

In our next example, we solve an application problem.

2. An abstract counting procedures adds one to a positive number and
subtracts one from the same number and finds the product of the two
numbers. If the product is 63, find the number.

Let x=the number. Then (x+1)(x−1) will be the product of one less
and one greater than the number. So our equation will be.

(x + 1)(x − 1) = 63

Multiplying we have :
x2 − 1 = 63

x2 = 64

Taking square roots :

Since the number must be positive, we have x = 8.

## 4 Facts

1. The general quadratic equation in standard form is written.

ax2 + bx + c = 0

2. To solve a quadratic equation simply write the equation in standard
form and substitute the coefficients into the general solution.

3. The solution to the quadratic equation is written:

## 5 Exercises

1. Use the quadratic formula to solve the equation.

t2 + 5t − 4 = 0

2. Use the quadratic formula to solve the equation.

5x2 + 7x − 2 = 0

3. Use the quadratic formula to solve the equation.

2x2 − 5x+2 = 0

4. In a group of children, each child gives a gift to every other child. If
the number of gifts is 132, find the number of children.

## 6 Solutions

1. Use the quadratic formula to solve the equation.

2. Use the quadratic formula to solve the equation.

3. Use the quadratic formula to solve the equation.

4. In a group of children, each child gives a gift to every other child. If
the number of gifts is 132, find the number of children.
Let x be the number of children. Since each child must give a gift to
every other child, each child must give x − 1 gifts. Then the following
equation will count the number of gifts.

Therefore, x = 12 or x = −11

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