Find practice problems and solutions for these concepts at Using the Quadratic Formula Practice Problems.
You have solved linear equations, radical equations, and quadratic equations using the factoring method. In this lesson, you will solve quadratic equations using the quadratic formula.
What Is a Quadratic Equation?
Previously, you solved quadratic equations using factoring. To refresh your memory, a quadratic equation is an equation whose highest exponent of the variable is 2. You might be asking yourself, "Why do I need to learn another method for solving quadratic equations when I already know how to solve them by using factoring? "Well, not all quadratic equations can be solved using factoring. You use the factoring method because it is faster and easier, but it will not always work. However, the quadratic formula, which is the method you will be using in this lesson, will always work.
A quadratic equation can be written in the form ax^{2} + bx + c = 0. The a represents the number in front of the x^{2} variable. The b represents the number in front of the x variable and c is the number. For instance, in the equation 2x^{2} + 3x + 5 = 0, the a is 2, the b is 3, and the c is 5. In the equation 4x^{2} – 6x + 7 = 0, the a is 4, the b is –6, and the c is 7. In the equation 5x^{2} + 7 = 0, the a is 5, the b is 0, and the c is 7. In the equation 8x^{2} – 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.
To use the quadratic formula, you need to know the a, b, and c of the equation. However, before you can determine what a, b, and c are, the equation must be in ax^{2} + bx+ c = 0 form. The equation 5x^{2} + 2x = 9 must be transformed to ax^{2} + bx + c = 0 form.
Example: 5x^{2} + 2x = 9  
Subtract 9 from both sides of the equation.  5x^{2} + 2x – 9 = 9 – 9 
Simplify.  5x^{2} + 2x – 9 = 0 
In this equation, a is 5, b is 2, and c is –9.
What Is the Quadratic Formula?
The quadratic formula is a formula that allows you to solve any quadratic equation—no matter how simple or difficult. If the equation is written ax^{2}+ bx+ c = 0, then the two solutions for x will be x = . It is the ± in the formula that gives us the two answers: one with + in that spot, and one with –. The formula contains a radical, which is one of the reasons you studied radicals in the previous lesson. To use the formula, you substitute the values of a, b, and c into the formula and then carry out the calculations.
Example: 3x^{2} – x – 2 = 0  
Determine a, b, and c.  a = 3, b = –1, and c = –2 
Take the quadratic formula.  
Substitute in the values of a, b, and c.  
Simplify.  
Simplify more.  
Take the square root of 25.  
The solutions are 1 and  and 
The origin of the quadratic formula is unknown, but its first recorded use was by the Babylonians in 400 BCE. They used the formula for everyday purposes—irrigating land, dividing money to pay workers, and mobilizing armies. 

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