SOLVING QUADRATIC EQUATIONS

Note:


Solve for x in the following equation.

Problem 4.1a tex2html_wrap_inline155 tex2html_wrap_inline185





Answer: tex2html_wrap_inline155 x = 2 ,5





Solution: tex2html_wrap_inline155 The equation is already equal to zero.





Method 1: tex2html_wrap_inline155 Factoring

The equation can be written in the equivalent form of tex2html_wrap_inline189 The only way that a product can equal zero is if one or both of the factors equal zero.

eqnarray47

The answers are tex2html_wrap_inline191





Method 2: tex2html_wrap_inline155 Completing the square

Subtract 10 from both sides of the equation.

eqnarray54

Add tex2html_wrap_inline193 to both sides of the equation.

eqnarray63

Factor the left side and simplify the right side.

eqnarray71

Take the square root of both sides of the equation,

eqnarray79

Add tex2html_wrap_inline195 to both sides of the equation.

eqnarray88





Method 3: tex2html_wrap_inline155 Quadratic Formula

The quadratic formula is tex2html_wrap_inline197

In the equation tex2html_wrap_inline185 , a is the coefficient of the tex2html_wrap_inline201 term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, -7 for b, and 10 for c in the quadratic formula and simplify.

eqnarray121





Method 4: tex2html_wrap_inline155 Graphing

Graph y= the left side of the equation or tex2html_wrap_inline213 and graph y= the right side of the equation or y=0. The graph of y=0 is nothing more than the x-axis. So what you will be looking for is where the graph of tex2html_wrap_inline213 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation. The x-intercepts are 2 and tex2html_wrap_inline227 The answers are 2 and tex2html_wrap_inline231





Check these answers in the original equation.

Check the answer x=2 by substituting 2 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 2 for x, then x=2 is a solution.





Check the solution x=5 by substituting 5 in the original equation for x. If the left side of the equation equals the right side of the equation after the substitution, you have found the correct answer.

Since the left side of the original equation is equal to the right side of the original equation after we substitute the value 5 for x, then x=5 is a solution.





The solutions to the equation tex2html_wrap_inline185 are 2 and 5.


If you would like to review the solution to problem 4.1b, click on Problem

If you would like to go back to the equation table of contents, click on Contents.

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Author:Nancy Marcus

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