quadratic equation

Solve quadratic equations by factoring. Solve quadratic equations by the square root property. Solve quadratic equations by completing the square. Solve quadratic equations by using the quadratic formula. Multiplying the factors expands the equation to a string of terms separated by plus or minus signs. So, in that sense, the operation of multiplication undoes the operation of factoring. For example, expand the factored expression (x−2)⁢(x+3) by multiplying the two factors together. We can use the zero-product property to solve quadratic equations in which we first have to factor out the greatest common factor(GCF), and for equations that have special factoring formulas as well, such as the difference of squares, both of which we will see later in this section. The last pair, 3⋅(−2) sums to 1, so these are the numbers. Note that only one pair of numbers will work. Then, write the factors. To solve this equation, we use the zero-product property. Set each factor equal to zero and solve. The two solutions are 2 and −3. We can see how the solutions relate to the graph in Figure 2.5.2. The solutions are the x-intercepts of x2+x−6 =0. Find two numbers whose product equals 15 and whose sum equals 8. List the factors of 15. The numbers that add to 8 are 3 and 5. Then, write the factors, set each factor equal to zero, and solve. Solve the difference of squares equation using the zero-product property: x2−9 =0. Recognizing that the equation represents the difference of squares, we can write the two factors by taking the square root of each term, using a minus sign as the operator in one factor and a plus sign as the operator in the other. Solve using the zero-factor property. When the leading coefficient is not 1, we factor a quadratic equation using the method called grouping , which requires four terms. The only pair of factors that sums to 15 is 3+12. Rewrite the equation replacing the b term, 15⁢x, with two terms using 3 and 12 as coefficients of x. Factor the first two terms, and then factor the last two terms. Example 2.5.6: Solving a Simple Quadratic Equation Using the Square Root Property Take the square root of both sides, and then simplify the radical. Remember to use a ± sign before the radical symbol. First, isolate the x2 term. Then take the square root of both sides. First, move the constant term to the right side of the equal sign. The fourth method of solving a quadratic equation is by using the quadratic formula , a formula that will solve all quadratic equations. Although the quadratic formula works on any quadratic equation in standard form, it is easy to make errors in substituting the values into the formula. Pay close attention when substituting, and use parentheses when inserting a negative number. Next, write the left side as a perfect square. Find the common denominator of the right side and write it as a single fraction: Finally, add −b2⁢a to both sides of the equation and combine the terms on the right side. Thus, The quadratic formula not only generates the solutions to a quadratic equation, it tells us about the nature of the solutions when we consider the discriminant , or the expression under the radical, b2−4⁢a⁢c. The discriminant tells us whether the solutions are real numbers or complex numbers, and how many solutions of each type to expect. Table 2.5.1 relates the value of the discriminant to the solutions of a quadratic equation. For a⁢x2+b⁢x+c =0, where a, b, and c are real numbers, the discriminant is the expression under the radical in the quadratic formula: b2−4⁢a⁢c. It tells us whether the solutions are real numbers or complex numbers and how many solutions of each type to expect. Example 2.5.11: Using the Discriminant to Find the Nature of the Solutions to a Quadratic Equation Calculate the discriminant b2−4⁢a⁢c for each equation and state the expected type of solutions. Example 2.5.12: Finding the Length of the Missing Side of a Right Triangle Find the length of the mis