Algebra Examples

y = x^{2} + 14 x + 21

$y = x^{2} + 14 x + 21$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

x^{2} + 14 x + 21

$x^{2} + 14 x + 21$ .

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Step 1.1.1

Use the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , to find the values of

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 1

$a = 1$

b = 14

$b = 14$

c = 21

$c = 21$

Step 1.1.2

Consider the vertex form of a parabola.

a {(x + d)}^{2} + e

$a {(x + d)}^{2} + e$

Step 1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

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Step 1.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

d = \frac{14}{2 \cdot 1}

$d = \frac{14}{2 \cdot 1}$

Step 1.1.3.2

Cancel the common factor of

14

$14$ and

2

$2$ .

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Step 1.1.3.2.1

Factor

2

$2$ out of

14

$14$ .

d = \frac{2 \cdot 7}{2 \cdot 1}

$d = \frac{2 \cdot 7}{2 \cdot 1}$

Step 1.1.3.2.2

Cancel the common factors.

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Step 1.1.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

d = \frac{2 \cdot 7}{2 (1)}

$d = \frac{2 \cdot 7}{2 (1)}$

Step 1.1.3.2.2.2

Cancel the common factor.

d = \frac{2 \cdot 7}{2 \cdot 1}

$d = \frac{2 \cdot 7}{2 \cdot 1}$

Step 1.1.3.2.2.3

Rewrite the expression.

d = \frac{7}{1}

$d = \frac{7}{1}$

Step 1.1.3.2.2.4

Divide

7

$7$ by

1

$1$ .

d = 7

$d = 7$

d = 7

$d = 7$

d = 7

$d = 7$

d = 7

$d = 7$

Step 1.1.4

Find the value of

e

$e$ using the formula

e = c - \frac{b^{2}}{4 a}

$e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

e = c - \frac{b^{2}}{4 a}

$e = c - \frac{b^{2}}{4 a}$ .

e = 21 - \frac{14^{2}}{4 \cdot 1}

$e = 21 - \frac{14^{2}}{4 \cdot 1}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Raise

14

$14$ to the power of

2

$2$ .

e = 21 - \frac{196}{4 \cdot 1}

$e = 21 - \frac{196}{4 \cdot 1}$

Step 1.1.4.2.1.2

Multiply

4

$4$ by

1

$1$ .

e = 21 - \frac{196}{4}

$e = 21 - \frac{196}{4}$

Step 1.1.4.2.1.3

Divide

196

$196$ by

4

$4$ .

e = 21 - 1 \cdot 49

$e = 21 - 1 \cdot 49$

Step 1.1.4.2.1.4

Multiply

- 1

$- 1$ by

49

$49$ .

e = 21 - 49

$e = 21 - 49$

e = 21 - 49

$e = 21 - 49$

Step 1.1.4.2.2

Subtract

49

$49$ from

21

$21$ .

e = - 28

$e = - 28$

e = - 28

$e = - 28$

e = - 28

$e = - 28$

Step 1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

{(x + 7)}^{2} - 28

${(x + 7)}^{2} - 28$ .

{(x + 7)}^{2} - 28

${(x + 7)}^{2} - 28$

{(x + 7)}^{2} - 28

${(x + 7)}^{2} - 28$

Step 1.2

Set

y

$y$ equal to the new right side.

y = {(x + 7)}^{2} - 28

$y = {(x + 7)}^{2} - 28$

y = {(x + 7)}^{2} - 28

$y = {(x + 7)}^{2} - 28$

Step 2

Use the vertex form,

y = a {(x - h)}^{2} + k

$y = a {(x - h)}^{2} + k$ , to determine the values of

a

$a$ ,

h

$h$ , and

k

$k$ .

a = 1

$a = 1$

h = - 7

$h = - 7$

k = - 28

$k = - 28$

Step 3

Find the vertex

(h, k)

$(h, k)$ .

(- 7, - 28)

$(- 7, - 28)$

Step 4

[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]

$[\begin{array}{c} x^{2} & \frac{1}{2} \\ \sqrt{π} & \int x d x \end{array}]$