Step: 1

The area of a right triangle is given by, A = 1 2 x base x height

Step: 2

48 = 1 2 x (x + 4) x x

[Substitute 48 for A, x for height and (x + 4) for base.]

Step: 3

48 x 2 = 1 2 x (x + 4) x x x 2

[Multiply each side by 2.]

Step: 4

96 = x (x + 4)

[Simplify.]

Step: 5

96 = x ^{2} + 4x

[Use distributive property.]

Step: 6

96 + 2^{2} = x ^{2} + 4x + 2^{2}

[To make RHS a perfect square, add (4/2)^{2} = (2)^{2} to each side.]

Step: 7

100 = (x + 2)^{2}

[Write right side as a perfect square.]

Step: 8

± 10 = (x + 2)

[Find square roots on each side.]

Step: 9

[Subtract 2 from each side.]

Step: 10

[Since x is the height of the triangle, discard the negative value.]

Correct Answer is : 8

Step: 1

[Original equation.]

Step: 2

[Add ( 8 2 )^{2} = 4^{2} to each side.]

Step: 3

(x + 4)^{2} = 49

[Writing left side as perfect square.]

Step: 4

(x + 4) = ± 7

[Finding square roots on each side.]

Step: 5

[Subtract 4 from each side.]

Step: 6

[Simplify.]

Step: 7

The solutions of the equation x ^{2} + 8x = 33 are - 11 and 3.

Correct Answer is : - 11 and 3

Step: 1

[Original equation.]

Step: 2

[Add (- 10 2 )^{2} = (- 5)^{2} = 25 to each
side.]

Step: 3

(x - 5)^{2} = 36

[Write left side as perfect square and simplify.]

Step: 4

[Evaluate square roots on both sides.]

Step: 5

[Add 5 to each side.]

Step: 6

[Simplify.]

Step: 7

The solutions of the equation x ^{2} - 10x = 11 are 11 and - 1.

Correct Answer is : 11 and - 1

Step: 1

- 2x ^{2} + 20x = - 18

Step: 2

[Divide throughout by - 2.]

Step: 3

[Add (- 10 2 )^{2} = (- 5)^{2} = 25 to both sides of the equation.]

Step: 4

(x - 5)^{2} = 34

[Write left side as perfect square and simplify.]

Step: 5

(x - 5) = ± 3 4

[Find the square root of both sides.]

Step: 6

[Add 5 to both sides of the equation.]

Step: 7

Step: 8

The solutions of the equation - 2x ^{2} + 20x = - 18 are x = 5 + 3 4 and x = 5 - 3 4 .

Correct Answer is : 5 ± 3 4

Step: 1

The area of a parallelogram is equal to the product of its height and length.

Step: 2

27 = x (x + 6)

[Original equation.]

Step: 3

27 = x ^{2} + 6x

[Use distributive property to simplify.]

Step: 4

27 + 3^{2} = x ^{2} + 6x + 3^{2}

[Add (6 2 )^{2} = (3)^{2} = 9 to each side.]

Step: 5

36 = (x + 3)^{2}

[Write right side as a perfect square and simplify.]

Step: 6

± 6 = (x + 3)

[Evaluate square roots on both sides.]

Step: 7

[Subtract 3 from each side.]

Step: 8

[Simplify.]

Step: 9

Height of the parallelogram is x = 3 cm.

[Since height cannot be a negative value.]

Step: 10

Length = x + 6 = 3 + 6 = 9 cm

[Simplify.]

Correct Answer is : 9 cm

Step: 1

[Given.]

Step: 2

[Subtract 3 from both sides.]

Step: 3

[Add (10 2 )^{2} or 5^{2} to each side.]

Step: 4

(x + 5)^{2} = 22

[Factor left side.]

Step: 5

Step: 6

Correct Answer is : - 5 ± 2 2

Step: 1

[Given.]

Step: 2

[Add 6 to both sides.]

Step: 3

[Add ( - 1 0 2 ) 2 or 5 2 to each side.]

Step: 4

(x - 5)^{2} = 31

Step: 5

Step: 6

Correct Answer is : 5 ± 3 1

Step: 1

[Original equation.]

Step: 2

[Subtracting 307 from the two sides of the equation.]

Step: 3

[Add (16 2 )^{2} = (8)^{2} = 64 to both sides of the equation.]

Step: 4

(x + 8)^{2} = - 243

[Write left side as perfect square and simplify.]

Step: 5

(x + 8) = ±( - 2 4 3 )

[Find the square root of both sides.]

Step: 6

(x + 8) = ± 93 i

[( - 2 4 3 ) = 2 4 3 i 2 = 93 i .]

Step: 7

[Subtracting 8 from the two sides of the equation.]

Step: 8

Step: 9

The solutions of the equation x ^{2} + 16x + 307 = 0 are x = - 8 + 93 i and x = - 8 - 93 i .

Correct Answer is : - 8 ± 93 i

Step: 1

- m ^{2} - 8m = 32

[Original equation.]

Step: 2

[Multiply throughout by - 1.]

Step: 3

[Add (8 2 )^{2} = (4)^{2} = 16 to both sides of the equation.]

Step: 4

(m + 4)^{2} = - 16

[Write left side as perfect square and simplify.]

Step: 5

(m + 4) = ± ( - 1 6 )

[Find the square root of both sides.]

Step: 6

(m + 4) = ± 4i

[( - 1 6 ) = ( 4 i ) 2 = 4i .]

Step: 7

[Subtracting 4 from the two sides of the equation.]

Step: 8

Step: 9

The solutions of the equation - m ^{2} - 8m = 32 are m = - 4 + 4i and m = - 4 - 4i .

Correct Answer is : - 4 ± 4i

Step: 1

[Given.]

Step: 2

[Add 15 to both sides.]

Step: 3

[Add ( - 6 2 )^{2} or 3 2 to each side.]

Step: 4

(x + 3)^{2} = 24

[Writing left side as perfect square.]

Step: 5

[Square root property.]

Step: 6

Step: 7

[Simplify.]

Correct Answer is : - 3 ± 2 6

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