# Section 8.1 Solving Quadratic Equations ax b c , with

```Section 8.1 Solving Quadratic Equations
A linear equation has the form ax  b  c , with a  0 . These equations are called first-degree
polynomial equations. In this section we learn how to solve second-degree polynomial
equations. These equations are called quadratic equations.
A quadratic equation in one variable is an equation that can be written in the form
ax2  bx  c  0 , where a , b , and c are real numbers, and a  0 . Quadratic equations in
this form are said to be in standard form.
Objective 1: Solve quadratic equations using factoring
Some quadratic equations can be solved quickly by factoring and by using the zero product
property introduced in Section 5.4.
Step 1. Write the quadratic equation in the standard form ax2  bx  c  0 .
Step 2. Factor the left-hand side.
Step 3. Set each factor from step 2 equal to zero (zero product property), and solve the
resulting linear equations.
Step 4. Check each potential solution in the original equation.
8.1.4 Solve the equation by factoring.
Objective 2: Solve quadratic equations using the square root property
Square Root Property
If u is an algebraic expression and k is a real number, then u 2  k is equivalent to u   k or
u  k . Equivalently, if u 2  k then u   k .
Solving Quadratic Equations Using the Square Root Property
Step 1. Write the equation in the form u 2  k to isolate the quantity being squared.
Step 2. Apply the square root property.
Step 3. Solve the resulting equations.
Step 4. Check the solutions in the original equation.
8.1.9 Solve the equation using the square root property.
8.1.11 Solve the equation using the square root property.
Objective 3: Solve quadratic equations by completing the square
Solving ax 2 + bx + c = 0 , a  0 , by Completing the Square
Step 1. If a  1 , divide both sides of the equation by a .
Step 2. Move all constants to the right-hand side.
Step 3. Find ½ times b (the coefficient of the x -term), square it, and add it to both sides of the
equation.
Step 4. The left-hand side is now a perfect square. Rewrite it as a binomial squared (i.e. factor it)
Step 5. Use the square root property and solve for x.
8.1.15 Decide what number must be added to make a perfect square trinomial.
8.1.17 Solve the quadratic equation by completing the square.
8.1.21 Solve the quadratic equation by completing the square.
Using the method of completing the square with a general quadratic equation
the quadratic formula is developed. Watch the video online to see how this is done.
The solutions to the quadratic equation ax2  bx  c  0 , a  0 , are given by the following
formula:
b  b 2  4ac
2a
CAUTION: Always write the variable for which you are solving (
x
)
,
CAUTION: On quizzes and exams you might not be directed to use a particular method.
Objective 5: Use the discriminant to determine the number and type of solutions to a
Discriminant
Given a quadratic equation ax2  bx  c  0 , a  0 , the expression D  b2  4ac is called the
discriminant.
If D  0 , then the quadratic equation has two real solutions.
If D  0 , then the quadratic equation has two non-real solutions.
If D  0 , then the quadratic equation has exactly one real solution.
8.1.29 Use the discriminant to determine the number and nature of the solutions of the quadratic
equation. Do not solve the equation.
8.1.31 Use the discriminant to determine the number and nature of the solutions of the quadratic
equation. Do not solve the equation.
Objective 6: Solve equations that are quadratic in form
Some equations can be changed into a quadratic equation using substitution.
8.1.32 Solve the equation after making an appropriate substitution.
8.1.33 Solve the equation after making an appropriate substitution.
8.1.34 Solve the equation after making an appropriate substitution.
8.1.38 Solve the equation after making an appropriate substitution.
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