Algebra Examples

$- 4 x^{2} + 56 x + 7 y - 203 = 0$

Step 1

Isolate $y$ to the left side of the equation.

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Step 1.1

Move all terms not containing $y$ to the right side of the equation.

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Step 1.1.1

Add $4 x^{2}$ to both sides of the equation.

$56 x + 7 y - 203 = 4 x^{2}$

Step 1.1.2

Subtract $56 x$ from both sides of the equation.

$7 y - 203 = 4 x^{2} - 56 x$

Step 1.1.3

Add $203$ to both sides of the equation.

$7 y = 4 x^{2} - 56 x + 203$

Step 1.2

Divide each term in $7 y = 4 x^{2} - 56 x + 203$ by $7$ and simplify.

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Step 1.2.1

Divide each term in $7 y = 4 x^{2} - 56 x + 203$ by $7$ .

$\frac{7 y}{7} = \frac{4 x^{2}}{7} + \frac{- 56 x}{7} + \frac{203}{7}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{7 y}{7} = \frac{4 x^{2}}{7} + \frac{- 56 x}{7} + \frac{203}{7}$

Step 1.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{4 x^{2}}{7} + \frac{- 56 x}{7} + \frac{203}{7}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Simplify each term.

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Step 1.2.3.1.1

Cancel the common factor of $- 56$ and $7$ .

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Step 1.2.3.1.1.1

Factor $7$ out of $- 56 x$ .

$y = \frac{4 x^{2}}{7} + \frac{7 (- 8 x)}{7} + \frac{203}{7}$

Step 1.2.3.1.1.2

Cancel the common factors.

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Step 1.2.3.1.1.2.1

Factor $7$ out of $7$ .

$y = \frac{4 x^{2}}{7} + \frac{7 (- 8 x)}{7 (1)} + \frac{203}{7}$

Step 1.2.3.1.1.2.2

Cancel the common factor.

$y = \frac{4 x^{2}}{7} + \frac{7 (- 8 x)}{7 \cdot 1} + \frac{203}{7}$

Step 1.2.3.1.1.2.3

Rewrite the expression.

$y = \frac{4 x^{2}}{7} + \frac{- 8 x}{1} + \frac{203}{7}$

Step 1.2.3.1.1.2.4

Divide $- 8 x$ by $1$ .

$y = \frac{4 x^{2}}{7} - 8 x + \frac{203}{7}$

Step 1.2.3.1.2

Divide $203$ by $7$ .

$y = \frac{4 x^{2}}{7} - 8 x + 29$

Step 2

Complete the square for $\frac{4 x^{2}}{7} - 8 x + 29$ .

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Step 2.1

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

$a = \frac{4}{7}$

$b = - 8$

$c = 29$

Step 2.2

Consider the vertex form of a parabola.

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

Step 2.3

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 2.3.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{- 8}{2 (\frac{4}{7})}$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Cancel the common factor of $- 8$ and $2$ .

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Step 2.3.2.1.1

Factor $2$ out of $- 8$ .

$d = \frac{2 \cdot - 4}{2 (\frac{4}{7})}$

Step 2.3.2.1.2

Cancel the common factors.

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Step 2.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot - 4}{2 (\frac{4}{7})}$

Step 2.3.2.1.2.2

Rewrite the expression.

$d = \frac{- 4}{\frac{4}{7}}$

Step 2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = - 4 (\frac{7}{4})$

Step 2.3.2.3

Cancel the common factor of $4$ .

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Step 2.3.2.3.1

Factor $4$ out of $- 4$ .

$d = 4 (- 1) \frac{7}{4}$

Step 2.3.2.3.2

Cancel the common factor.

$d = 4 \cdot - 1 \frac{7}{4}$

Step 2.3.2.3.3

Rewrite the expression.

$d = - 1 \cdot 7$

Step 2.3.2.4

Multiply $- 1$ by $7$ .

$d = - 7$

Step 2.4

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 2.4.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 29 - \frac{{(- 8)}^{2}}{4 (\frac{4}{7})}$

Step 2.4.2

Simplify the right side.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Raise $- 8$ to the power of $2$ .

$e = 29 - \frac{64}{4 (\frac{4}{7})}$

Step 2.4.2.1.2

Combine $4$ and $\frac{4}{7}$ .

$e = 29 - \frac{64}{\frac{4 \cdot 4}{7}}$

Step 2.4.2.1.3

Multiply $4$ by $4$ .

$e = 29 - \frac{64}{\frac{16}{7}}$

Step 2.4.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = 29 - (64 (\frac{7}{16}))$

Step 2.4.2.1.5

Cancel the common factor of $16$ .

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Step 2.4.2.1.5.1

Factor $16$ out of $64$ .

$e = 29 - (16 (4) \frac{7}{16})$

Step 2.4.2.1.5.2

Cancel the common factor.

$e = 29 - (16 \cdot 4 \frac{7}{16})$

Step 2.4.2.1.5.3

Rewrite the expression.

$e = 29 - (4 \cdot 7)$

Step 2.4.2.1.6

Multiply $4$ by $7$ .

$e = 29 - 1 \cdot 28$

Step 2.4.2.1.7

Multiply $- 1$ by $28$ .

$e = 29 - 28$

Step 2.4.2.2

Subtract $28$ from $29$ .

$e = 1$

Step 2.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{4}{7} {(x - 7)}^{2} + 1$ .

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

Step 3

Set $y$ equal to the new right side.

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