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Solve by completing the square.

-2x2+5x+1=0

AdamTaurus  Aug 31, 2017

Best Answer 

 #1
avatar+1088 
+2

I actually do not like the method of completing the square because I believe it to be quite inefficient, but I'll use it, if you insist.

 

Using the method of completing the square, I will solve for x in the equation \(-2x^2+5x+1=0\).

 

\(-2x^2+5x+1=0\) In a quadratic equation in the form, \(ax^2+bx+c=0\) we must move c to the other side; in other words, subtract 1 on both sides of the equation.
\(-2x^2+5x=-1\) Before we can proceed, we must make the coefficient of the quadratic term 1. At the moment, it is -2. Therefore, we must divide by -2 on both sides.
\(x^2-\frac{5}{2}x=\frac{1}{2}\) Now, the next part is probably the hardest to understand. Our goal in the next step is make \(x^2-\frac{5}{2}x\) a perfect-square trinomial by adding a value. In order to do that, we must add \(\left(\frac{b}{2}\right)^2\) where b is the coefficient of the linear term. 
\(x^2-\frac{5}{2}x+\left(\frac{b}{2}\right)^2=\frac{1}{2}+\left(\frac{b}{2}\right)^2\) I added (b/2)^2 to both sides because whatever you do one side, you must do to the other in order to keep the equations balanced. Now, plug in the appropriate value of b, (-5/2) in this case.
\(x^2-\frac{5}{2}x+\left(\frac{\frac{-5}{2}}{2}\right)^2=\frac{1}{2}+\left(\frac{\frac{-5}{2}}{2}\right)^2\) Now, I will simplify within the parentheses. 
\(\frac{\left(\frac{-5}{2}\right)}{2}=\frac{-5}{2}\div\frac{2}{1}=\frac{-5}{2}*\frac{1}{2}=\frac{-5}{4}\) Reinsert this.
\(x^2-\frac{5}{2}x+\left(\frac{-5}{4}\right)^2=\frac{1}{2}+\left(\frac{-5}{4}\right)^2\) Distribute the exponent to both the numerator and denominator.
\(x^2-\frac{5}{2}x+\frac{(-5)^2}{4^2}=\frac{1}{2}+\frac{(-5)^2}{4^2}\) Simplify both the numerator and the denominator of both fractions.
\(x^2-\frac{5}{2}x+\frac{25}{16}=\frac{1}{2}+\frac{25}{16}\) Let's simplify the right hand side of the equation. In this case, this requires converting 1/2 into a fraction with the denominator of 16.
\(\frac{1}{2}+\frac{25}{16}=\frac{8}{16}+\frac{25}{16}=\frac{33}{16}\)  
\(x^2-\frac{5}{2}x+\frac{25}{16}=\frac{33}{16}\) By adding (b/2)^2 to both sides of the equation, we have created a perfect-square trinomial on the left-hand side. It may not be obvious because of the fractions, but it may help to know that the trinomial, when manipulated correctly, will always become in the form of \(\left(x+\frac{b}{2}\right)^2\)
\(\left(x-\frac{5}{4}\right)^2=\frac{33}{16}\) Take the square root of both sides. Remember that the square root of both sides results in both the absolute value.
\(\left|x-\frac{5}{4}\right|=\sqrt{\frac{33}{16}}\) Let's simplify the right hand side of the equation by distributing the square root to both the numerator and the denominator.
\(\left|x-\frac{5}{4}\right|={\frac{\sqrt{33}}{\sqrt{16}}}\) The square root of 16 is 4, so let's simplify that.
\(\left|x-\frac{5}{4}\right|=\frac{\sqrt{33}}{4}\) Now, solve for both positive and negative answer.
\(x-\frac{5}{4}=\frac{\sqrt{33}}{4}\) \(-\left(x-\frac{5}{4}\right)=\frac{\sqrt{33}}{4}\)

 

 
\(x=\frac{\sqrt{33}}{4}+\frac{5}{4}\) \(-x+\frac{5}{4}=\frac{\sqrt{33}}{4}\)

 

 
\(x_1=\frac{\sqrt{33}+5}{4}\) \(-x=\frac{\sqrt{33}}{4}-\frac{5}{4}\)

 

 
\(x_1=\frac{\sqrt{33}+5}{4}\) \(-x=\frac{\sqrt{33}-5}{4}\)

 

 
\(x_1=\frac{\sqrt{33}+5}{4}\) \(x_2=\frac{-\sqrt{33}+5}{4}\)

 

 
TheXSquaredFactor  Aug 31, 2017
Sort: 

1+0 Answers

 #1
avatar+1088 
+2
Best Answer

I actually do not like the method of completing the square because I believe it to be quite inefficient, but I'll use it, if you insist.

 

Using the method of completing the square, I will solve for x in the equation \(-2x^2+5x+1=0\).

 

\(-2x^2+5x+1=0\) In a quadratic equation in the form, \(ax^2+bx+c=0\) we must move c to the other side; in other words, subtract 1 on both sides of the equation.
\(-2x^2+5x=-1\) Before we can proceed, we must make the coefficient of the quadratic term 1. At the moment, it is -2. Therefore, we must divide by -2 on both sides.
\(x^2-\frac{5}{2}x=\frac{1}{2}\) Now, the next part is probably the hardest to understand. Our goal in the next step is make \(x^2-\frac{5}{2}x\) a perfect-square trinomial by adding a value. In order to do that, we must add \(\left(\frac{b}{2}\right)^2\) where b is the coefficient of the linear term. 
\(x^2-\frac{5}{2}x+\left(\frac{b}{2}\right)^2=\frac{1}{2}+\left(\frac{b}{2}\right)^2\) I added (b/2)^2 to both sides because whatever you do one side, you must do to the other in order to keep the equations balanced. Now, plug in the appropriate value of b, (-5/2) in this case.
\(x^2-\frac{5}{2}x+\left(\frac{\frac{-5}{2}}{2}\right)^2=\frac{1}{2}+\left(\frac{\frac{-5}{2}}{2}\right)^2\) Now, I will simplify within the parentheses. 
\(\frac{\left(\frac{-5}{2}\right)}{2}=\frac{-5}{2}\div\frac{2}{1}=\frac{-5}{2}*\frac{1}{2}=\frac{-5}{4}\) Reinsert this.
\(x^2-\frac{5}{2}x+\left(\frac{-5}{4}\right)^2=\frac{1}{2}+\left(\frac{-5}{4}\right)^2\) Distribute the exponent to both the numerator and denominator.
\(x^2-\frac{5}{2}x+\frac{(-5)^2}{4^2}=\frac{1}{2}+\frac{(-5)^2}{4^2}\) Simplify both the numerator and the denominator of both fractions.
\(x^2-\frac{5}{2}x+\frac{25}{16}=\frac{1}{2}+\frac{25}{16}\) Let's simplify the right hand side of the equation. In this case, this requires converting 1/2 into a fraction with the denominator of 16.
\(\frac{1}{2}+\frac{25}{16}=\frac{8}{16}+\frac{25}{16}=\frac{33}{16}\)  
\(x^2-\frac{5}{2}x+\frac{25}{16}=\frac{33}{16}\) By adding (b/2)^2 to both sides of the equation, we have created a perfect-square trinomial on the left-hand side. It may not be obvious because of the fractions, but it may help to know that the trinomial, when manipulated correctly, will always become in the form of \(\left(x+\frac{b}{2}\right)^2\)
\(\left(x-\frac{5}{4}\right)^2=\frac{33}{16}\) Take the square root of both sides. Remember that the square root of both sides results in both the absolute value.
\(\left|x-\frac{5}{4}\right|=\sqrt{\frac{33}{16}}\) Let's simplify the right hand side of the equation by distributing the square root to both the numerator and the denominator.
\(\left|x-\frac{5}{4}\right|={\frac{\sqrt{33}}{\sqrt{16}}}\) The square root of 16 is 4, so let's simplify that.
\(\left|x-\frac{5}{4}\right|=\frac{\sqrt{33}}{4}\) Now, solve for both positive and negative answer.
\(x-\frac{5}{4}=\frac{\sqrt{33}}{4}\) \(-\left(x-\frac{5}{4}\right)=\frac{\sqrt{33}}{4}\)

 

 
\(x=\frac{\sqrt{33}}{4}+\frac{5}{4}\) \(-x+\frac{5}{4}=\frac{\sqrt{33}}{4}\)

 

 
\(x_1=\frac{\sqrt{33}+5}{4}\) \(-x=\frac{\sqrt{33}}{4}-\frac{5}{4}\)

 

 
\(x_1=\frac{\sqrt{33}+5}{4}\) \(-x=\frac{\sqrt{33}-5}{4}\)

 

 
\(x_1=\frac{\sqrt{33}+5}{4}\) \(x_2=\frac{-\sqrt{33}+5}{4}\)

 

 
TheXSquaredFactor  Aug 31, 2017

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