Quadratic Equations and Functions Lesson 4 Quadratic Equations and Functions

Overview We have been dealing with linear equations till now. A linear equation has variables whose highest power is one. Linear equations with one variable have one solution. Linear equations with two variables may have a unique solution, no solution or infinite solutions. We also know that the graph of a linear equation is a straight line. Now we shall deal with other types of equations

Quadratic Expression Definition Let us now move on to equations which contain higher powers of their variables. Let us see the given product, (x + 1) (x + 2) = x2 + 3x + 2 Here the highest power of the variable is 2. Such expression whose highest power of the variable (degree) is 2 are called quadratic expressions. Example: 1) y2 + 4y - 2 2) x2 - 3x + 6

Quadratic Euation Definition Any equation which, after possible rearrangement or regrouping of terms can be written in the form ax2 + bx + c = 0, where a, b, c are real numbers and a ≠ 0,  is called a quadratic equation in x. (If a = 0 then it becomes a linear equation as x2 term becomes 0) ax2 + bx + c = 0 is called the standard form of the quadratic equation. As the equation contains only one unknown x, it is an equation with one variable. (Remember: a, b and c are constants.)

Binomials Product Let us see how a quadratic expression is formed. Let us multiply two binomials namely (x + 2)(x + 3) using distributive property (x + 2)(x + 3) = x(x + 3) + 2(x + 3) = x2 + 3x + 2x + 6 = x2 + 5x + 6

Binomials Let us take another example, (x - 1)(x + 4)  = x(x + 4) - 1(x + 4) = x2 + 4x - x - 4 = x2 + 3x - 4 Therefore, we can conclude that the product of two binomials whose power of the variable is one is a quadratic expression.

Factorization Splitting the Middle Term As we have seen that a quadratic expression is the product of two binomials, a quadratic expression can be written as the product of two binomials. It is easy to find the product, but not so to split a quadratic expression into its component factors.

Factorization There is a method to factorize a quadratic expression known as splitting the middle term. We shall study in detail about it. We know (x + 3)(x + 4) = x2 + 3x + 4x + 12 = x2 + (3 + 4)x + 3 * 4 = x2 + 7x + 12 Here we observe that the coefficient of the middle term is the sum of the two numbers in each linear binomial and the last term is the product of the two numbers in each binomial. Example: 3x + 2

Solution of a QE Explanation A quadratic equation is of the form ax2 + bx + c = 0, where a ≠ 0. Factorization of a quadratic expression helps in solving a quadratic equation. Let us see how quadratic equations can be solved. Solve: x2 + 5x - 36 = 0

Solution of a QE Solution: Here we observe that the given equation does not match any of the given identities some will use the method of splitting the middle term. x2 + 5x - 36 = 0 We must choose two numbers such that a + b = 5, a * b = -36 so 9 - 4 = 5 and 9 * (-4) = -36

Quadratic Formula Formula Sometimes we cannot factorize and solve a quadratic equation using the methods mentioned such as splitting the middle term or using standard identities, then we use the quadratic formula. The quadratic equation will not have real roots if b2 - 4ac is negative and we write no solution in the set of real numbers. If b2 - 4ac = 0, then both roots and are equal. If b2 - 4ac > 0, then the roots are real and distinct.

Functions Introduction We come across the term functions in mathematics very often that shows how important they are. We shall see what they are and how they are useful.

Functions If there are 20 students in a class, then the number of seats required is 20. If there are 25 students naturally 25 seats are required, but we don’t say that as there are 50 seats we will have 50 students in a class. It is clear that the number of seats in a class room depends on the number of students but the number of students does not depend upon the number of seats. Hence we say the number of seats in a class is a function of the number of students in the class.

References Online Free SAT Study Guide: SAT Guide http://www.proprofs.com/sat/study- guide/index.shtml