Use Completing the Square to Derive the Quadratic Formula: CCSS.Math.Content.HSA-REI.B.4a - Common Core: High School - Algebra

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Question

Solve $x^{2} + 17 x - 33 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 17 x - 33 + 33 = 0 + 33$

$x^{2} + 17 x = 33$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 17 x + \frac{17}{2} ^2 = 33 + \frac{17}{2} ^2$

$x^{2} + 17 x + \frac{289}{4} = 33 + \frac{289}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{17}{2} \right)^2= \frac{421}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{17}{2} \right)^2}= \sqrt{ \frac{421}{4} }$

$x + \frac{17}{2} = \sqrt{ \frac{421}{4} }$

Now we subtract $\frac{17}{2}$ from each side.

$x + \frac{17}{2} - \frac{17}{2} = \sqrt{ \frac{421}{4} }- \frac{17}{2}$

$x = - \frac{17}{2} + \frac{\sqrt{421}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{17}{2} + \frac{\sqrt{421}}{2}$

$x = - \frac{\sqrt{421}}{2} - \frac{17}{2}$

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Question

Solve the following by completing the square. Round your answer to the nearest hundredth:

$x^{2} + 19 x - 52 = 0$

Answer

The first step is to add to both sides.

$x^{2} + 19 x - 52 + 52 = 0 + 52$

$x^{2} + 19 x = 52$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 19 x + \frac{19}{2} ^2 = 52 + \frac{19}{2} ^2$

$x^{2} + 19 x + \frac{361}{4} = 52 + \frac{361}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{19}{2} \right)^2= \frac{569}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{19}{2} \right)^2}= \sqrt{ \frac{569}{4} }$

$x + \frac{19}{2} = \sqrt{ \frac{569}{4} }$

Now we subtract $\frac{19}{2}$ from each side.

$x + \frac{19}{2} - \frac{19}{2} = \sqrt{ \frac{569}{4} }- \frac{19}{2}$

$x = - \frac{19}{2} + \frac{\sqrt{569}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{19}{2} + \frac{\sqrt{569}}{2}$

$x = - \frac{\sqrt{569}}{2} - \frac{19}{2}$

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Question

Solve $x^{2} + 21 x - 84 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 21 x - 84 + 84 = 0 + 84$

$x^{2} + 21 x = 84$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 21 x + \frac{21}{2} ^2 = 84 + \frac{21}{2} ^2$

$x^{2} + 21 x + \frac{441}{4} = 84 + \frac{441}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{21}{2} \right)^2= \frac{777}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{21}{2} \right)^2}= \sqrt{ \frac{777}{4} }$

$x + \frac{21}{2} = \sqrt{ \frac{777}{4} }$

Now we subtract $\frac{21}{2}$ from each side.

$x + \frac{21}{2} - \frac{21}{2} = \sqrt{ \frac{777}{4} }- \frac{21}{2}$

$x = - \frac{21}{2} + \frac{\sqrt{777}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{21}{2} + \frac{\sqrt{777}}{2}$

$x = - \frac{\sqrt{777}}{2} - \frac{21}{2}$

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Question

Solve $x^{2} + 39 x - 77 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 39 x - 77 + 77 = 0 + 77$

$x^{2} + 39 x = 77$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 39 x + \frac{39}{2} ^2 = 77 + \frac{39}{2} ^2$

$x^{2} + 39 x + \frac{1521}{4} = 77 + \frac{1521}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{39}{2} \right)^2= \frac{1829}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{39}{2} \right)^2}= \sqrt{ \frac{1829}{4} }$

$x + \frac{39}{2} = \sqrt{ \frac{1829}{4} }$

Now we subtract $\frac{39}{2}$ from each side.

$x + \frac{39}{2} - \frac{39}{2} = \sqrt{ \frac{1829}{4} }- \frac{39}{2}$

$x = - \frac{39}{2} + \frac{\sqrt{1829}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{39}{2} + \frac{\sqrt{1829}}{2}$

$x = - \frac{\sqrt{1829}}{2} - \frac{39}{2}$

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Question

Solve $x^{2} + 11 x - 67 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 11 x - 67 + 67 = 0 + 67$

$x^{2} + 11 x = 67$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 11 x + \frac{11}{2} ^2 = 67 + \frac{11}{2} ^2$

$x^{2} + 11 x + \frac{121}{4} = 67 + \frac{121}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{11}{2} \right)^2= \frac{389}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{11}{2} \right)^2}= \sqrt{ \frac{389}{4} }$

$x + \frac{11}{2} = \sqrt{ \frac{389}{4} }$

Now we subtract $\frac{11}{2}$ from each side.

$x + \frac{11}{2} - \frac{11}{2} = \sqrt{ \frac{389}{4} }- \frac{11}{2}$

$x = - \frac{11}{2} + \frac{\sqrt{389}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{11}{2} + \frac{\sqrt{389}}{2}$

$x = - \frac{\sqrt{389}}{2} - \frac{11}{2}$

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Question

Solve $x^{2} + 41 x - 19 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 41 x - 19 + 19 = 0 + 19$

$x^{2} + 41 x = 19$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 41 x + \frac{41}{2} ^2 = 19 + \frac{41}{2} ^2$

$x^{2} + 41 x + \frac{1681}{4} = 19 + \frac{1681}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{41}{2} \right)^2= \frac{1757}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{41}{2} \right)^2}= \sqrt{ \frac{1757}{4} }$

$x + \frac{41}{2} = \sqrt{ \frac{1757}{4} }$

Now we subtract $\frac{41}{2}$ from each side.

$x + \frac{41}{2} - \frac{41}{2} = \sqrt{ \frac{1757}{4} }- \frac{41}{2}$

$x = - \frac{41}{2} + \frac{\sqrt{1757}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{41}{2} + \frac{\sqrt{1757}}{2}$

$x = - \frac{\sqrt{1757}}{2} - \frac{41}{2}$

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Question

Solve $x^{2} + 57 x - 87 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 57 x - 87 + 87 = 0 + 87$

$x^{2} + 57 x = 87$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 57 x + \frac{57}{2} ^2 = 87 + \frac{57}{2} ^2$

$x^{2} + 57 x + \frac{3249}{4} = 87 + \frac{3249}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{57}{2} \right)^2= \frac{3597}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{57}{2} \right)^2}= \sqrt{ \frac{3597}{4} }$

$x + \frac{57}{2} = \sqrt{ \frac{3597}{4} }$

Now we subtract $\frac{57}{2}$ from each side.

$x + \frac{57}{2} - \frac{57}{2} = \sqrt{ \frac{3597}{4} }- \frac{57}{2}$

$x = - \frac{57}{2} + \frac{\sqrt{3597}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{57}{2} + \frac{\sqrt{3597}}{2}$

$x = - \frac{\sqrt{3597}}{2} - \frac{57}{2}$

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Question

Solve $x^{2} + 29 x - 45 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 29 x - 45 + 45 = 0 + 45$

$x^{2} + 29 x = 45$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 29 x + \frac{29}{2} ^2 = 45 + \frac{29}{2} ^2$

$x^{2} + 29 x + \frac{841}{4} = 45 + \frac{841}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{29}{2} \right)^2= \frac{1021}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{29}{2} \right)^2}= \sqrt{ \frac{1021}{4} }$

$x + \frac{29}{2} = \sqrt{ \frac{1021}{4} }$

Now we subtract $\frac{29}{2}$ from each side.

$x + \frac{29}{2} - \frac{29}{2} = \sqrt{ \frac{1021}{4} }- \frac{29}{2}$

$x = - \frac{29}{2} + \frac{\sqrt{1021}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{29}{2} + \frac{\sqrt{1021}}{2}$

$x = - \frac{\sqrt{1021}}{2} - \frac{29}{2}$

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Question

Solve $x^{2} + 25 x - 40 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add 40 to both sides.

$x^{2} + 25 x - 40 + 40 = 0 + 40$

$x^{2} + 25 x = 40$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 25 x + \frac{25}{2} ^2 = 40 + \frac{25}{2} ^2$

$x^{2} + 25 x + \frac{625}{4} = 40 + \frac{625}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{25}{2} \right)^2= \frac{785}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{25}{2} \right)^2}= \sqrt{ \frac{785}{4} }$

$x + \frac{25}{2} = \sqrt{ \frac{785}{4} }$

Now we subtract $\frac{25}{2}$ from each side.

$x + \frac{25}{2} - \frac{25}{2} = \sqrt{ \frac{785}{4} }- \frac{25}{2}$

$x = - \frac{25}{2} + \frac{\sqrt{785}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{25}{2} + \frac{\sqrt{785}}{2}$

$x = - \frac{\sqrt{785}}{2} - \frac{25}{2}$

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Question

Solve $x^{2} + 19 x - 2 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 19 x - 2 + 2 = 0 + 2$

$x^{2} + 19 x = 2$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 19 x + \frac{19}{2} ^2 = 2 + \frac{19}{2} ^2$

$x^{2} + 19 x + \frac{361}{4} = 2 + \frac{361}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{19}{2} \right)^2= \frac{369}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{19}{2} \right)^2}= \sqrt{ \frac{369}{4} }$

$x + \frac{19}{2} = \sqrt{ \frac{369}{4} }$

Now we subtract $\frac{19}{2}$ from each side.

$x + \frac{19}{2} - \frac{19}{2} = \sqrt{ \frac{369}{4} }- \frac{19}{2}$

$x = - \frac{19}{2} + \frac{3 \sqrt{41}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{19}{2} + \frac{3 \sqrt{41}}{2}$

$x = - \frac{3 \sqrt{41}}{2} - \frac{19}{2}$

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Question

Solve $x^{2} + 43 x - 19 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 43 x - 19 + 19 = 0 + 19$

$x^{2} + 43 x = 19$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 43 x + \frac{43}{2} ^2 = 19 + \frac{43}{2} ^2$

$x^{2} + 43 x + \frac{1849}{4} = 19 + \frac{1849}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{43}{2} \right)^2= \frac{1925}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{43}{2} \right)^2}= \sqrt{ \frac{1925}{4} }$

$x + \frac{43}{2} = \sqrt{ \frac{1925}{4} }$

Now we subtract $\frac{43}{2}$ from each side.

$x + \frac{43}{2} - \frac{43}{2} = \sqrt{ \frac{1925}{4} }- \frac{43}{2}$

$x = - \frac{43}{2} + \frac{5 \sqrt{77}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{43}{2} + \frac{5 \sqrt{77}}{2}$

$x = - \frac{5 \sqrt{77}}{2} - \frac{43}{2}$

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Question

Solve $x^{2} + 19 x - 34 = 0$ by completing the square. Round your answer to the nearest hundredth.

Answer

The first step is to add to both sides.

$x^{2} + 19 x - 34 + 34 = 0 + 34$

$x^{2} + 19 x = 34$

Now we take the coefficient in front of the $\uptext{x}$ term, divide it by , square it and add it to each side.

$x^{2} + 19 x + \frac{19}{2} ^2 = 34 + \frac{19}{2} ^2$

$x^{2} + 19 x + \frac{361}{4} = 34 + \frac{361}{4}$

Now we factor the left hand side, and add up the right hand side.

$\left( x + \frac{19}{2} \right)^2= \frac{497}{4}$

Now we take the square root of each side.

$\sqrt{\left( x + \frac{19}{2} \right)^2}= \sqrt{ \frac{497}{4} }$

$x + \frac{19}{2} = \sqrt{ \frac{497}{4} }$

Now we subtract $\frac{19}{2}$ from each side.

$x + \frac{19}{2} - \frac{19}{2} = \sqrt{ \frac{497}{4} }- \frac{19}{2}$

$x = - \frac{19}{2} + \frac{\sqrt{497}}{2}$

Since we are taking the square root, we need to set up equations to solve for $\uptext{x}$ .

$x = - \frac{19}{2} + \frac{\sqrt{497}}{2}$

$x = - \frac{\sqrt{497}}{2} - \frac{19}{2}$

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