SOLVING QUADRATIC EQUATIONS



Note:




Solve for x in the following equation.


Problem 4.3d:text2html_wrap_inline253tex2html_wrap_inline553



Answer:text2html_wrap_inline253tex2html_wrap_inline555 exact answers;

tex2html_wrap_inline557 and tex2html_wrap_inline559 approximate answers



Solution:

The equation is already set to zero.

Rewrite the original equation as an equivalent equation without denominators by multiplying both sides by 16.


eqnarray45



Simplify the left sides of the equation by combining like terms.


eqnarray57







Method 1:text2html_wrap_inline253Factoring

The equation tex2html_wrap_inline561 is not easily factored, so we will not use this method.







Method 2:text2html_wrap_inline253Completing the square

Add 173 to both sides of the equation tex2html_wrap_inline561 .


eqnarray73



Divide both sides by 128:


eqnarray87



Add tex2html_wrap_inline565 to both sides of the equation:


eqnarray114



Factor the left side and simplify the right side:


eqnarray136



Take the square root of both sides of the equation:


eqnarray154



Add tex2html_wrap_inline567 to both sides of the equation:


eqnarray170


eqnarray177







Method 3:text2html_wrap_inline253Quadratic Formula

The quadratic formula is tex2html_wrap_inline569

In the equation ,a is the coefficient of the tex2html_wrap_inline573 term, b is the coefficient of the x term, and c is the constant. Substitute 128 for a , -204 for b, and -173 for c in the quadratic formula and simplify.



eqnarray208


eqnarray215







Method 4:text2html_wrap_inline253Graphing

Graph tex2html_wrap_inline583 (formed by subtracting the right side of the original equation from the left side of the original equation). Graph y=0 (the x-axis). What you will be looking for is where the graph of tex2html_wrap_inline583 crosses the x-axis. Another way of saying this is that the x-intercepts are the solutions to this equation.

You can see from the graph that there are two x-intercepts located at 2.20633314611 and -0.612583146106. This means that there are two real answers: x=2.20633314611 and tex2html_wrap_inline597

The answers are 2.20633314611 and -0.612583146106. These answers may or may not be solutions to the original equation. You must check the answers with the original equation.



Check these answers in the original equation.



Check the solution x=2.20633314611 by substituting 2.20633314611 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 2.20633314611 for x, then x=2.20633314611 is a solution.



Check the solution x=-0.612583146106 by substituting -0.612583146106 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 -0.612583146106 for x, then x=-0.612583146106 is a solution.







The solutions to the equation tex2html_wrap_inline553 are

eqnarray285







Comment:text2html_wrap_inline253You can use the exact solutions to factor the left side of the


eqnarray293


For example, since tex2html_wrap_inline635 , then

eqnarray309


Since tex2html_wrap_inline641 , then

eqnarray322


Since the product tex2html_wrap_inline645


and tex2html_wrap_inline649


eqnarray354


displaymath551


However not the only factors:


Since the first term of the product tex2html_wrap_inline651 is not tex2html_wrap_inline653 there must be another factor of 8:


Let tex2html_wrap_inline657 s check to see whether tex2html_wrap_inline659


eqnarray411


eqnarray422


eqnarray441


eqnarray462


Therefore 8x tex2html_wrap_inline661 - tex2html_wrap_inline663 x- tex2html_wrap_inline665 is factored as 8 tex2html_wrap_inline667








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Author: Nancy Marcus

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