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

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Example Questions

Example Question #1 : Use Completing The Square To Derive The Quadratic Formula: Ccss.Math.Content.Hsa Rei.B.4a

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

Possible Answers:

x = 1.26, x = -18.26

x = 1.76 , x = -18.76

x = -0.74, x = -16.26

x = -6.76, x = 4.76

x = -0.52, x = -18.64

Correct answer:

x = 1.76 , x = -18.76

Explanation:

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 = 1.76

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

x = -18.76

Report an Error

Example Question #2 : Use Completing The Square To Derive The Quadratic Formula: Ccss.Math.Content.Hsa Rei.B.4a

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

Possible Answers:

x = -39.12, x = 14.09

x = 0.94, x = -21.94

x = 3.44 , x = -24.44

x = -6.28, x = 4.12

x = 2.94, x = -23.94

Correct answer:

x = 3.44 , x = -24.44

Explanation:

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 = 3.44

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

x = -24.44

Report an Error

Example Question #3 : Use Completing The Square To Derive The Quadratic Formula: Ccss.Math.Content.Hsa Rei.B.4a

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

Possible Answers:

x = -4.05, x = 4.28

x = 1.38, x = -40.38

x = -0.62, x = -38.38

x = 1.88 , x = -40.88

x = -49.47, x = -7.44

Correct answer:

x = 1.88 , x = -40.88

Explanation:

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 = 1.88

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

x = -40.88

Report an Error

Example Question #43 : Reasoning With Equations & Inequalities

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

Possible Answers:

x = 3.86, x = -14.86

x = 5.42, x = -8.34

x = 1.86, x = -12.86

x = 4.36 , x = -15.36

x = -32.32, x = 13.82

Correct answer:

x = 4.36 , x = -15.36

Explanation:

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 = 4.36

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

x = -15.36

Report an Error

Example Question #44 : Reasoning With Equations & Inequalities

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

Possible Answers:

x = -2.04, x = -38.96

x = -0.04, x = -40.96

x = 0.46 , x = -41.46

x = 19.33, x = -7.84

x = -2.03, x = 4.45

Correct answer:

x = 0.46 , x = -41.46

Explanation:

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 = 0.46

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

x = -41.46

Report an Error

Example Question #45 : Reasoning With Equations & Inequalities

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

Possible Answers:

x = -27.31, x = -2.9

x = 1.49 , x = -58.49

x = 4.16, x = -6.5

x = -1.01, x = -55.99

x = 0.99, x = -57.99

Correct answer:

x = 1.49 , x = -58.49

Explanation:

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 = 1.49

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

x = -58.49

Report an Error

Example Question #46 : Reasoning With Equations & Inequalities

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

Possible Answers:

x = 3.69, x = -6.08

x = 0.98, x = -29.98

x = -29.41, x = 47.98

x = -1.02, x = -27.98

x = 1.48 , x = -30.48

Correct answer:

x = 1.48 , x = -30.48

Explanation:

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 = 1.48

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

x = -30.48

Report an Error

Example Question #47 : Reasoning With Equations & Inequalities

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

Possible Answers:

x = -9.39, x = 8.59

x = 14.02, x = 45.3

x = 1.01, x = -26.01

x = 1.51 , x = -26.51

x = -0.99, x = -24.01

Correct answer:

x = 1.51 , x = -26.51

Explanation:

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 = 1.51

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

x = -26.51

Report an Error

Example Question #48 : Reasoning With Equations & Inequalities

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

Possible Answers:

x = -7.01, x = 5.94

x = 0.1 , x = -19.1

x = -6.66, x = -6.47

x = -2.4, x = -16.6

x = -0.4, x = -18.6

Correct answer:

x = 0.1 , x = -19.1

Explanation:

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 = 0.1

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

x = -19.1

Report an Error

Example Question #49 : Reasoning With Equations & Inequalities

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

Possible Answers:

x = -0.06, x = -42.94

x = 7.69, x = -8.09

x = -2.06, x = -40.94

x = 19.15, x = 30.43

x = 0.44 , x = -43.44

Correct answer:

x = 0.44 , x = -43.44

Explanation:

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 = 0.44

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

x = -43.44

Report an Error

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

Study concepts, example questions & explanations for Common Core: High School - Algebra

All Common Core: High School - Algebra Resources

Example Questions

Example Question #1 : Use Completing The Square To Derive The Quadratic Formula: Ccss.Math.Content.Hsa Rei.B.4a

Example Question #2 : Use Completing The Square To Derive The Quadratic Formula: Ccss.Math.Content.Hsa Rei.B.4a

Example Question #3 : Use Completing The Square To Derive The Quadratic Formula: Ccss.Math.Content.Hsa Rei.B.4a

Example Question #43 : Reasoning With Equations & Inequalities

Example Question #44 : Reasoning With Equations & Inequalities

Example Question #45 : Reasoning With Equations & Inequalities

Example Question #46 : Reasoning With Equations & Inequalities

Example Question #47 : Reasoning With Equations & Inequalities

Example Question #48 : Reasoning With Equations & Inequalities

Example Question #49 : Reasoning With Equations & Inequalities

All Common Core: High School - Algebra Resources

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