Algebra Examples

$y = - 2 x + \frac{21}{4}$ $y = x^{2} - x - \frac{3}{4}$

Step 1

Eliminate the equal sides of each equation and combine.

$- 2 x + \frac{21}{4} = x^{2} - x - \frac{3}{4}$

Step 2

Solve $- 2 x + \frac{21}{4} = x^{2} - x - \frac{3}{4}$ for $x$ .

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Step 2.1

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - x - \frac{3}{4} = - 2 x + \frac{21}{4}$

Step 2.2

Move all terms containing $x$ to the left side of the equation.

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Step 2.2.1

Add $2 x$ to both sides of the equation.

$x^{2} - x - \frac{3}{4} + 2 x = \frac{21}{4}$

Step 2.2.2

Add $- x$ and $2 x$ .

$x^{2} + x - \frac{3}{4} = \frac{21}{4}$

Step 2.3

Subtract $\frac{21}{4}$ from both sides of the equation.

$x^{2} + x - \frac{3}{4} - \frac{21}{4} = 0$

Step 2.4

Simplify $x^{2} + x - \frac{3}{4} - \frac{21}{4}$ .

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Step 2.4.1

Combine the numerators over the common denominator.

$x^{2} + x + \frac{- 3 - 21}{4} = 0$

Step 2.4.2

Subtract $21$ from $- 3$ .

$x^{2} + x + \frac{- 24}{4} = 0$

Step 2.4.3

Divide $- 24$ by $4$ .

$x^{2} + x - 6 = 0$

Step 2.5

Factor $x^{2} + x - 6$ using the AC method.

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Step 2.5.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $1$ .

$- 2, 3 + y = x^{2} - x - \frac{3}{4}$

Step 2.5.2

Write the factored form using these integers.

$(x - 2) (x + 3) = 0$

Step 2.6

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 2 = 0$

$x + 3 = 0 + y = x^{2} - x - \frac{3}{4}$

Step 2.7

Set $x - 2$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $x - 2$ equal to $0$ .

$x - 2 = 0$

Step 2.7.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.8

Set $x + 3$ equal to $0$ and solve for $x$ .

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Step 2.8.1

Set $x + 3$ equal to $0$ .

$x + 3 = 0$

Step 2.8.2

Subtract $3$ from both sides of the equation.

$x = - 3$

Step 2.9

The final solution is all the values that make $(x - 2) (x + 3) = 0$ true.

$x = 2, - 3$

Step 3

Evaluate $y$ when $x = 2$ .

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Step 3.1

Substitute $2$ for $x$ .

$y = {(2)}^{2} - (2) - \frac{3}{4}$

Step 3.2

Substitute $2$ for $x$ in $y = {(2)}^{2} - (2) - \frac{3}{4}$ and solve for $y$ .

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Step 3.2.1

Remove parentheses.

$y = 2^{2} - (2) - \frac{3}{4}$

Step 3.2.2

Simplify $2^{2} - (2) - \frac{3}{4}$ .

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Step 3.2.2.1

Find the common denominator.

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Step 3.2.2.1.1

Write $2^{2}$ as a fraction with denominator $1$ .

$y = \frac{2^{2}}{1} - (2) - \frac{3}{4}$

Step 3.2.2.1.2

Multiply $\frac{2^{2}}{1}$ by $\frac{4}{4}$ .

$y = \frac{2^{2}}{1} \cdot \frac{4}{4} - (2) - \frac{3}{4}$

Step 3.2.2.1.3

Multiply $\frac{2^{2}}{1}$ by $\frac{4}{4}$ .

$y = \frac{2^{2} \cdot 4}{4} - (2) - \frac{3}{4}$

Step 3.2.2.1.4

Write $- (2)$ as a fraction with denominator $1$ .

$y = \frac{2^{2} \cdot 4}{4} + \frac{- (2)}{1} - \frac{3}{4}$

Step 3.2.2.1.5

Multiply $\frac{- (2)}{1}$ by $\frac{4}{4}$ .

$y = \frac{2^{2} \cdot 4}{4} + \frac{- (2)}{1} \cdot \frac{4}{4} - \frac{3}{4}$

Step 3.2.2.1.6

Multiply $\frac{- (2)}{1}$ by $\frac{4}{4}$ .

$y = \frac{2^{2} \cdot 4}{4} + \frac{- (2) \cdot 4}{4} - \frac{3}{4}$

Step 3.2.2.2

Combine the numerators over the common denominator.

$y = \frac{2^{2} \cdot 4 - (2) \cdot 4 - 3}{4}$

Step 3.2.2.3

Simplify each term.

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Step 3.2.2.3.1

Raise $2$ to the power of $2$ .

$y = \frac{4 \cdot 4 - (2) \cdot 4 - 3}{4}$

Step 3.2.2.3.2

Multiply $4$ by $4$ .

$y = \frac{16 - (2) \cdot 4 - 3}{4}$

Step 3.2.2.3.3

Multiply $- (2) \cdot 4$ .

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Step 3.2.2.3.3.1

Multiply $- 1$ by $2$ .

$y = \frac{16 - 2 \cdot 4 - 3}{4}$

Step 3.2.2.3.3.2

Multiply $- 2$ by $4$ .

$y = \frac{16 - 8 - 3}{4}$

Step 3.2.2.4

Simplify by subtracting numbers.

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Step 3.2.2.4.1

Subtract $8$ from $16$ .

$y = \frac{8 - 3}{4}$

Step 3.2.2.4.2

Subtract $3$ from $8$ .

$y = \frac{5}{4}$

Step 4

Evaluate $y$ when $x = - 3$ .

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Step 4.1

Substitute $- 3$ for $x$ .

$y = {(- 3)}^{2} - (- 3) - \frac{3}{4}$

Step 4.2

Simplify ${(- 3)}^{2} - (- 3) - \frac{3}{4}$ .

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Step 4.2.1

Find the common denominator.

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Step 4.2.1.1

Write ${(- 3)}^{2}$ as a fraction with denominator $1$ .

$y = \frac{{(- 3)}^{2}}{1} - (- 3) - \frac{3}{4}$

Step 4.2.1.2

Multiply $\frac{{(- 3)}^{2}}{1}$ by $\frac{4}{4}$ .

$y = \frac{{(- 3)}^{2}}{1} \cdot \frac{4}{4} - (- 3) - \frac{3}{4}$

Step 4.2.1.3

Multiply $\frac{{(- 3)}^{2}}{1}$ by $\frac{4}{4}$ .

$y = \frac{{(- 3)}^{2} \cdot 4}{4} - (- 3) - \frac{3}{4}$

Step 4.2.1.4

Write $- (- 3)$ as a fraction with denominator $1$ .

$y = \frac{{(- 3)}^{2} \cdot 4}{4} + \frac{- (- 3)}{1} - \frac{3}{4}$

Step 4.2.1.5

Multiply $\frac{- (- 3)}{1}$ by $\frac{4}{4}$ .

$y = \frac{{(- 3)}^{2} \cdot 4}{4} + \frac{- (- 3)}{1} \cdot \frac{4}{4} - \frac{3}{4}$

Step 4.2.1.6

Multiply $\frac{- (- 3)}{1}$ by $\frac{4}{4}$ .

$y = \frac{{(- 3)}^{2} \cdot 4}{4} + \frac{- (- 3) \cdot 4}{4} - \frac{3}{4}$

Step 4.2.2

Combine the numerators over the common denominator.

$y = \frac{{(- 3)}^{2} \cdot 4 - (- 3) \cdot 4 - 3}{4}$

Step 4.2.3

Simplify each term.

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Step 4.2.3.1

Raise $- 3$ to the power of $2$ .

$y = \frac{9 \cdot 4 - (- 3) \cdot 4 - 3}{4}$

Step 4.2.3.2

Multiply $9$ by $4$ .

$y = \frac{36 - (- 3) \cdot 4 - 3}{4}$

Step 4.2.3.3

Multiply $- (- 3) \cdot 4$ .

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Step 4.2.3.3.1

Multiply $- 1$ by $- 3$ .

$y = \frac{36 + 3 \cdot 4 - 3}{4}$

Step 4.2.3.3.2

Multiply $3$ by $4$ .

$y = \frac{36 + 12 - 3}{4}$

Step 4.2.4

Simplify by adding and subtracting.

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Step 4.2.4.1

Add $36$ and $12$ .

$y = \frac{48 - 3}{4}$

Step 4.2.4.2

Subtract $3$ from $48$ .

$y = \frac{45}{4}$

Step 5

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(2, \frac{5}{4})$

$(- 3, \frac{45}{4})$

Step 6

The result can be shown in multiple forms.

Point Form:

$(2, \frac{5}{4}), (- 3, \frac{45}{4})$

Equation Form:

$x = 2, y = \frac{5}{4}$

$x = - 3, y = \frac{45}{4}$

Step 7