SOLVING QUADRATIC EQUATIONS

Note:


Solve for x in the following equation.

Problem 4.1c tex2html_wrap_inline155 tex2html_wrap_inline603





Answer: tex2html_wrap_inline155 tex2html_wrap_inline605





Solution: tex2html_wrap_inline155 The equation is already equal to zero.





Method 1: tex2html_wrap_inline155 Factoring

The equation cannot be easily factored.





Method 2: tex2html_wrap_inline155 Completing the square

Subtract 10 from both sides of the equation.

eqnarray405

Add tex2html_wrap_inline607 to both sides of the equation.

eqnarray415

Factor the left side and simplify the right side.

eqnarray423

Take the square root of both sides of the equation,

eqnarray431

Subtract tex2html_wrap_inline609 from both sides of the equation.

eqnarray440

and

eqnarray450





Method 3: tex2html_wrap_inline155 Quadratic Formula

The quadratic formula is tex2html_wrap_inline611

In the equation tex2html_wrap_inline603 , a is the coefficient of the tex2html_wrap_inline615 term, b is the coefficient of the x term, and c is the constant. Simply insert 1 for a, +17 for b, and +10 for c in the quadratic formula and simplify.

eqnarray469





Method 4: tex2html_wrap_inline155 Graphing

Graph y = the left side of the equation or tex2html_wrap_inline627 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_inline627 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation. The x-intercepts are -16.389866919 and tex2html_wrap_inline641 The answers are -16.389866919 and tex2html_wrap_inline645





Check these answers in the original equation.

Check the answer x=-16.389866919 by substituting -16.389866919 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 -16.389866919 for x, then x=-16.389866919 is a solution.





Check the solution x=-0.61013308097 by substituting -0.61013308097 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 -0.61013308097 for x, then x=-0.61013308097 is a solution.





The solutions to the equation tex2html_wrap_inline603 are , and .





Comment: tex2html_wrap_inline155 You can use the solutions to factor the original equation.

For example, since tex2html_wrap_inline681 , then tex2html_wrap_inline683 , and tex2html_wrap_inline685

Since tex2html_wrap_inline687 , then tex2html_wrap_inline689 , and tex2html_wrap_inline691

Since the product tex2html_wrap_inline693 and tex2html_wrap_inline603 , then we can say that tex2html_wrap_inline697 This means that tex2html_wrap_inline699 and tex2html_wrap_inline701 are factors of tex2html_wrap_inline703


If you would like to review the solution to problem 4.1d, 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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