Algebra Examples

x^{2} - 7 x + 10 = 0

$x^{2} - 7 x + 10 = 0$

Step 1

Subtract

10

$10$ from both sides of the equation.

x^{2} - 7 x = - 10

$x^{2} - 7 x = - 10$

Step 2

To create a trinomial square on the left side of the equation, find a value that is equal to the square of half of

b

$b$ .

{(\frac{b}{2})}^{2} = {(- \frac{7}{2})}^{2}

${(\frac{b}{2})}^{2} = {(- \frac{7}{2})}^{2}$

Step 3

Add the term to each side of the equation.

x^{2} - 7 x + {(- \frac{7}{2})}^{2} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + {(- \frac{7}{2})}^{2} = - 10 + {(- \frac{7}{2})}^{2}$

Step 4

Simplify the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify each term.

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Step 4.1.1.1

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.1.1.1.1

Apply the product rule to

- \frac{7}{2}

$- \frac{7}{2}$ .

x^{2} - 7 x + {(- 1)}^{2} {(\frac{7}{2})}^{2} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + {(- 1)}^{2} {(\frac{7}{2})}^{2} = - 10 + {(- \frac{7}{2})}^{2}$

Step 4.1.1.1.2

Apply the product rule to

\frac{7}{2}

$\frac{7}{2}$ .

x^{2} - 7 x + {(- 1)}^{2} \frac{7^{2}}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + {(- 1)}^{2} \frac{7^{2}}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}$

x^{2} - 7 x + {(- 1)}^{2} \frac{7^{2}}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + {(- 1)}^{2} \frac{7^{2}}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}$

Step 4.1.1.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

x^{2} - 7 x + 1 \frac{7^{2}}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + 1 \frac{7^{2}}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}$

Step 4.1.1.3

Multiply

\frac{7^{2}}{2^{2}}

$\frac{7^{2}}{2^{2}}$ by

1

$1$ .

x^{2} - 7 x + \frac{7^{2}}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + \frac{7^{2}}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}$

Step 4.1.1.4

Raise

7

$7$ to the power of

2

$2$ .

x^{2} - 7 x + \frac{49}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + \frac{49}{2^{2}} = - 10 + {(- \frac{7}{2})}^{2}$

Step 4.1.1.5

Raise

2

$2$ to the power of

2

$2$ .

x^{2} - 7 x + \frac{49}{4} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + \frac{49}{4} = - 10 + {(- \frac{7}{2})}^{2}$

x^{2} - 7 x + \frac{49}{4} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + \frac{49}{4} = - 10 + {(- \frac{7}{2})}^{2}$

x^{2} - 7 x + \frac{49}{4} = - 10 + {(- \frac{7}{2})}^{2}

$x^{2} - 7 x + \frac{49}{4} = - 10 + {(- \frac{7}{2})}^{2}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify

- 10 + {(- \frac{7}{2})}^{2}

$- 10 + {(- \frac{7}{2})}^{2}$ .

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Step 4.2.1.1

Simplify each term.

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Step 4.2.1.1.1

Use the power rule

{(a b)}^{n} = a^{n} b^{n}

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 4.2.1.1.1.1

Apply the product rule to

- \frac{7}{2}

$- \frac{7}{2}$ .

x^{2} - 7 x + \frac{49}{4} = - 10 + {(- 1)}^{2} {(\frac{7}{2})}^{2}

$x^{2} - 7 x + \frac{49}{4} = - 10 + {(- 1)}^{2} {(\frac{7}{2})}^{2}$

Step 4.2.1.1.1.2

Apply the product rule to

\frac{7}{2}

$\frac{7}{2}$ .

x^{2} - 7 x + \frac{49}{4} = - 10 + {(- 1)}^{2} \frac{7^{2}}{2^{2}}

$x^{2} - 7 x + \frac{49}{4} = - 10 + {(- 1)}^{2} \frac{7^{2}}{2^{2}}$

x^{2} - 7 x + \frac{49}{4} = - 10 + {(- 1)}^{2} \frac{7^{2}}{2^{2}}

$x^{2} - 7 x + \frac{49}{4} = - 10 + {(- 1)}^{2} \frac{7^{2}}{2^{2}}$

Step 4.2.1.1.2

Raise

- 1

$- 1$ to the power of

2

$2$ .

x^{2} - 7 x + \frac{49}{4} = - 10 + 1 \frac{7^{2}}{2^{2}}

$x^{2} - 7 x + \frac{49}{4} = - 10 + 1 \frac{7^{2}}{2^{2}}$

Step 4.2.1.1.3

Multiply

\frac{7^{2}}{2^{2}}

$\frac{7^{2}}{2^{2}}$ by

1

$1$ .

x^{2} - 7 x + \frac{49}{4} = - 10 + \frac{7^{2}}{2^{2}}

$x^{2} - 7 x + \frac{49}{4} = - 10 + \frac{7^{2}}{2^{2}}$

Step 4.2.1.1.4

Raise

7

$7$ to the power of

2

$2$ .

x^{2} - 7 x + \frac{49}{4} = - 10 + \frac{49}{2^{2}}

$x^{2} - 7 x + \frac{49}{4} = - 10 + \frac{49}{2^{2}}$

Step 4.2.1.1.5

Raise

2

$2$ to the power of

2

$2$ .

x^{2} - 7 x + \frac{49}{4} = - 10 + \frac{49}{4}

$x^{2} - 7 x + \frac{49}{4} = - 10 + \frac{49}{4}$

x^{2} - 7 x + \frac{49}{4} = - 10 + \frac{49}{4}

$x^{2} - 7 x + \frac{49}{4} = - 10 + \frac{49}{4}$

Step 4.2.1.2

To write

- 10

$- 10$ as a fraction with a common denominator, multiply by

\frac{4}{4}

$\frac{4}{4}$ .

x^{2} - 7 x + \frac{49}{4} = - 10 \cdot \frac{4}{4} + \frac{49}{4}

$x^{2} - 7 x + \frac{49}{4} = - 10 \cdot \frac{4}{4} + \frac{49}{4}$

Step 4.2.1.3

Combine

- 10

$- 10$ and

\frac{4}{4}

$\frac{4}{4}$ .

x^{2} - 7 x + \frac{49}{4} = \frac{- 10 \cdot 4}{4} + \frac{49}{4}

$x^{2} - 7 x + \frac{49}{4} = \frac{- 10 \cdot 4}{4} + \frac{49}{4}$

Step 4.2.1.4

Combine the numerators over the common denominator.

x^{2} - 7 x + \frac{49}{4} = \frac{- 10 \cdot 4 + 49}{4}

$x^{2} - 7 x + \frac{49}{4} = \frac{- 10 \cdot 4 + 49}{4}$

Step 4.2.1.5

Simplify the numerator.

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Step 4.2.1.5.1

Multiply

- 10

$- 10$ by

4

$4$ .

x^{2} - 7 x + \frac{49}{4} = \frac{- 40 + 49}{4}

$x^{2} - 7 x + \frac{49}{4} = \frac{- 40 + 49}{4}$

Step 4.2.1.5.2

Add

- 40

$- 40$ and

49

$49$ .

x^{2} - 7 x + \frac{49}{4} = \frac{9}{4}