Quadratic Trinomials, Quadratic Equations, and the Quadratic Formula Help
Introduction to Qudratic Trinomials and Equations
A quadratic trinomial contains an x2 term as well as an x term. For example, x2 – 6x + 8 is a quadratic trinomial. You can factor quadratic trinomials by using the FOIL method in reverse.
Let's factor x2 – 6x + 8.
Start by looking at the last term in the trinomial: 8. Ask yourself, "What two integers, when multiplied together, have a product of positive 8?" Make a mental list of these integers:
1 ×8 –1× –8 2 ×4 –2× –4
Next look at the middle term of the trinomial: –6x. Choose the two factors from the list you just made that also add up to the coefficient –6:
–2 and –4
Now write the factors using –2 and –4:
(x – 2)(x – 4)
Use the FOIL method to double-check your answer:
(x – 2)(x – 4) = x2 – 6x + 8
You can see that the answer is correct.
A quadratic equation is an equation that does not graph into a straight line. The graph will be a smooth curve. An equation is a quadratic equation if the highest exponent of the variable is 2. Here are some examples of quadratic equations:
x2 + 6x + 10 = 0
6x2 + 8x – 22 = 0
A quadratic equation can be written in the form: ax2+ bx + c = 0. The a represents the number in front of the x2 variable. The b represents the number in front of the x variable and c is the number. For instance, in the equation 2x2 + 3x + 5 = 0, the a is 2, the b is 3, and the c is 5. In the equation 4x2 – 6x + 7 = 0, the a is 4, the b is –6, and the c is 7. In the equation 5x2 + 7 = 0, the a is 5, the b is 0, and the c is 7. In the equation 8x2 – 3x = 0, the a is 8, the b is –3, and the c is 0. Is the equation 2x + 7 = 0 a quadratic equation? No! The equation does not contain a variable with an exponent of 2. Therefore, it is not a quadratic equation.
Solving Quadratic Equations Using Factoring
Why is the equation x2 = 4 a quadratic equation? It is a quadratic equation because the variable has an exponent of 2. To solve a quadratic equation, first make one side of the equation zero. Let's work with x2 = 4.
Subtract 4 from both sides of the equation to make one side of the equation zero: x2 – 4 = 4 – 4. Now, simplify x2 – 4 = 0. The next step is to factor x2 – 4. It can be factored as the difference of two squares: (x – 2)(x + 2) = 0.
If ab = 0, you know that either a or b or both factors have to be zero because a times b = 0. This is called the zero product property, and it says that if the product of two numbers is zero, then one or both of the numbers have to be zero. You can use this idea to help solve quadratic equations with the factoring method.
Use the zero product property, and set each factor equal to zero: (x – 2) = 0 and (x + 2) = 0.
When you use the zero product property, you get linear equations that you already know how to solve.
Solve the equation: x – 2 = 0 Add 2 to both sides of the equation. x – 2 + 2 = 0 + 2 Now, simplify: x = 2 Solve the equation: x + 2 = 0 Subtract 2 from both sides of the equation. x + 2 – 2 = 0 – 2 Simplify: x = –2
You got two values for x. The two solutions for x are 2 and –2. All quadratic equations have two solutions. The exponent 2 in the equation tells you that the equation is quadratic, and it also tells you that you will have two answers.
Tip: When both your solutions are the same number, this is called a double root. You will get a double root when both factors are the same.
Before you can factor an expression, the expression must be arranged in descending order. An expression is in descending order when you start with the largest exponent and descend to the smallest, as shown in this example: 2x2 + 5x + 6 = 0.
All quadratic equations have two solutions. The exponent of 2 in the equation tells you to expect two answers.
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