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. Quadratic Equation in One Variable 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. Solving Quadratic Equations by Factoring 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. Objective 4: Solve quadratic equations using the quadratic formula 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. Quadratic Formula 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 ) , 8.1.24 Solve the quadratic equation using the quadratic formula. 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 quadratic equation. 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.