In Algebra, we often encounter equations that involve variables raised to the second power. These types of equations are called Quadratic Equations. A quadratic equation is an equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable we are solving for. The term "quadratic" comes from the Latin word "quadratus," which means "square." This is because the variable x is squared in these types of equations. Quadratic equations can be used to solve real-world problems, such as calculating the maximum height of a ball thrown into the air or determining the dimensions of a rectangular garden. To solve a quadratic equation, there are several methods we can use, including factoring, completing the square, and using the quadratic formula. In this course, we will explore each of these methods in detail. It is important to note that not all equations with variables raised to the second power are quadratic equations. For example, an equation like x^2 + 3x - 5 = x^3 is not a quadratic equation because it contains a variable raised to the third power. In the next section, we will learn how to identify quadratic equations and how to solve them using factoring.To find the roots of a quadratic equation, we need to solve for the values of x that make the equation equal to zero. The roots of a quadratic equation are also referred to as its zeros, solutions or x-intercepts. There are different methods to find the roots of a quadratic equation, but one of the most common is the quadratic formula. The quadratic formula applies to any quadratic equation in the form ax^2 + bx + c = 0, where a, b, and c are constants.The quadratic formula is given by:x = (-b ± √(b^2 - 4ac)) / 2aTo use the quadratic formula, we simply substitute the values of a, b, and c into the formula and solve for x. Note that the expression inside the square root symbol is called the discriminant, and its value determines the nature of the roots.- If the discriminant is positive, then the quadratic equation has two distinct real roots.- If the discriminant is zero, then the quadratic equation has one real root with a multiplicity of two (also known as a double root).- If the discriminant is negative, then the quadratic equation has two complex conjugate roots.Let's look at an example:Solve the quadratic equation 2x^2 + 5x - 3 = 0 using the quadratic formula.First, we identify the values of a, b, and c:a = 2, b = 5, c = -3Then, we substitute these values into the quadratic formula and simplify:x = (-5 ± √(5^2 - 4(2)(-3))) / 2(2)x = (-5 ± √49) / 4x = (-5 ± 7) / 4Therefore, the roots of the quadratic equation 2x^2 + 5x - 3 = 0 are:x = (-5 + 7) / 4 = 1/2x = (-5 - 7) / 4 = -3/2So, the equation has two distinct real roots.Remember, finding the roots of a quadratic equation is an essential skill in solving problems that involve quadratic functions.

Now that we have learned how to solve quadratic equations using the Quadratic Formula, let's explore another method called factoring. Factoring is the process of breaking down a quadratic equation into two separate linear expressions.

The general form of a quadratic equation is ax² + bx + c = 0, where a, b, and c are constants. To factor a quadratic equation, we need to find two binomials that multiply together to equal the original quadratic equation.

Let's take the quadratic equation x² + 5x + 6 = 0 as an example. To factor this equation, we need to find two binomials in the form (x + ?)(x + ?) that multiply together to equal x² + 5x + 6.

To do this, we need to find two numbers that multiply together to equal 6 and add up to 5. The two numbers are 2 and 3. So, we can factor the quadratic equation as (x + 2)(x + 3) = 0.

Now, we can use the Zero Product Property, which states that if the product of two factors is equal to zero, then at least one of the factors must be equal to zero. So, we set each factor equal to zero and solve for x.

(x + 2)(x + 3) = 0

x + 2 = 0 or x + 3 = 0

x = -2 or x = -3

Therefore, the solutions to the quadratic equation x² + 5x + 6 = 0 are x = -2 and x = -3.

Note that not all quadratic equations can be factored using integer coefficients. In such cases, we need to use the Quadratic Formula to solve for the solutions.

Practice factoring quadratic equations by trying the problems in the exercises section below.

Completing the Square:Completing the square is an algebraic technique that is used to convert a quadratic equation into a perfect square trinomial form. The method is useful in solving quadratic equations and graphing quadratic functions. The following steps are used to complete the square:Step 1: Write the quadratic equation in standard form: ax² + bx + c = 0Step 2: Move the constant term to the right-hand side of the equation: ax² + bx = -cStep 3: Divide both sides of the equation by "a" to make the coefficient of x² equal to 1: x² + (b/a)x = -c/aStep 4: Take half of the coefficient of x and square it: (b/2a)²Step 5: Add the result obtained in step 4 to both sides of the equation: x² + (b/a)x + (b/2a)² = (b/2a)² - c/aStep 6: Simplify the equation: (x + (b/2a))² = (b² - 4ac)/4a²Step 7: Take the square root of both sides: x + (b/2a) = ±√((b² - 4ac)/4a²)Step 8: Simplify: x = (-b ± √(b² - 4ac))/2aExample: Solve the quadratic equation x² + 6x + 5 = 0 by completing the square.Step 1: Write the equation in standard form: x² + 6x + 5 = 0Step 2: Move the constant term to the right-hand side of the equation: x² + 6x = -5Step 3: Divide both sides of the equation by "1" to make the coefficient of x² equal to 1: x² + 6x = -5Step 4: Take half of the coefficient of x and square it: (6/2)² = 9Step 5: Add 9 to both sides of the equation: x² + 6x + 9 = 4Step 6: Simplify the equation: (x + 3)² = 4Step 7: Take the square root of both sides: x + 3 = ±2Step 8: Simplify: x = -3 ± 2Therefore, the solutions to the quadratic equation x² + 6x + 5 = 0 are x = -1 and x = -5.Completing the square is a useful technique that helps in the solutions of quadratic equations. With practice, you will find it easier to use and understand.

Now that we have learned how to solve quadratic equations by factoring, we will learn another method known as the quadratic formula. The quadratic formula is a formula that can be used to find the solutions of any quadratic equation. It is given as:

where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0. The quadratic formula is derived by completing the square on the general quadratic equation.

Let's try an example:

Example:

Solve the quadratic equation 2x^2 + 5x - 3 = 0 using the quadratic formula.

Solution:

Here, a = 2, b = 5 and c = -3. Substituting these values in the quadratic formula, we get:

Simplifying the expression inside the square root, we get:

Therefore, the solutions of the quadratic equation 2x^2 + 5x - 3 = 0 are:

So, the solutions are x = 1/2 or x = -3/2.

The quadratic formula is a powerful tool for solving quadratic equations, especially when factoring is difficult or not possible. However, it is important to remember that it is always a good idea to check your solutions by plugging them back into the original equation and verifying that they satisfy it.