Algebra Examples

$2 y = 2 x^{2} + 4 x - 8$ $y = 2 x^{2} - 28$

Step 1

Replace all occurrences of $y$ with $2 x^{2} - 28$ in each equation.

Tap for more steps...

Step 1.1

Replace all occurrences of $y$ in $2 y = 2 x^{2} + 4 x - 8$ with $2 x^{2} - 28$ .

$2 (2 x^{2} - 28) = 2 x^{2} + 4 x - 8$

$y = 2 x^{2} - 28$

Step 1.2

Simplify the left side.

Tap for more steps...

Step 1.2.1

Simplify $2 (2 x^{2} - 28)$ .

Tap for more steps...

Step 1.2.1.1

Apply the distributive property.

$2 (2 x^{2}) + 2 \cdot - 28 = 2 x^{2} + 4 x - 8$

$y = 2 x^{2} - 28$

Step 1.2.1.2

Multiply.

Tap for more steps...

Step 1.2.1.2.1

Multiply $2$ by $2$ .

$4 x^{2} + 2 \cdot - 28 = 2 x^{2} + 4 x - 8$

$y = 2 x^{2} - 28$

Step 1.2.1.2.2

Multiply $2$ by $- 28$ .

$4 x^{2} - 56 = 2 x^{2} + 4 x - 8$

$y = 2 x^{2} - 28$

$4 x^{2} - 56 = 2 x^{2} + 4 x - 8$

$y = 2 x^{2} - 28$

$4 x^{2} - 56 = 2 x^{2} + 4 x - 8$

$y = 2 x^{2} - 28$

$4 x^{2} - 56 = 2 x^{2} + 4 x - 8$

$y = 2 x^{2} - 28$

$4 x^{2} - 56 = 2 x^{2} + 4 x - 8$

$y = 2 x^{2} - 28$

Step 2

Solve for $x$ in $4 x^{2} - 56 = 2 x^{2} + 4 x - 8$ .

Tap for more steps...

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.

$2 x^{2} + 4 x - 8 = 4 x^{2} - 56$

$y = 2 x^{2} - 28$

Step 2.2

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

Tap for more steps...

Step 2.2.1

Subtract $4 x^{2}$ from both sides of the equation.

$2 x^{2} + 4 x - 8 - 4 x^{2} = - 56$

$y = 2 x^{2} - 28$

Step 2.2.2

Subtract $4 x^{2}$ from $2 x^{2}$ .

$- 2 x^{2} + 4 x - 8 = - 56$

$y = 2 x^{2} - 28$

$- 2 x^{2} + 4 x - 8 = - 56$

$y = 2 x^{2} - 28$

Step 2.3

Add $56$ to both sides of the equation.

$- 2 x^{2} + 4 x - 8 + 56 = 0$

$y = 2 x^{2} - 28$

Step 2.4

Add $- 8$ and $56$ .

$- 2 x^{2} + 4 x + 48 = 0$

$y = 2 x^{2} - 28$

Step 2.5

Factor the left side of the equation.

Tap for more steps...

Step 2.5.1

Factor $- 2$ out of $- 2 x^{2} + 4 x + 48$ .

Tap for more steps...

Step 2.5.1.1

Factor $- 2$ out of $- 2 x^{2}$ .

$- 2 x^{2} + 4 x + 48 = 0$

$y = 2 x^{2} - 28$

Step 2.5.1.2

Factor $- 2$ out of $4 x$ .

$- 2 x^{2} - 2 (- 2 x) + 48 = 0$

$y = 2 x^{2} - 28$

Step 2.5.1.3

Factor $- 2$ out of $48$ .

$- 2 x^{2} - 2 (- 2 x) - 2 \cdot - 24 = 0$

$y = 2 x^{2} - 28$

Step 2.5.1.4

Factor $- 2$ out of $- 2 (x^{2}) - 2 (- 2 x)$ .

$- 2 (x^{2} - 2 x) - 2 \cdot - 24 = 0$

$y = 2 x^{2} - 28$

Step 2.5.1.5

Factor $- 2$ out of $- 2 (x^{2} - 2 x) - 2 (- 24)$ .

$- 2 (x^{2} - 2 x - 24) = 0$

$y = 2 x^{2} - 28$

$- 2 (x^{2} - 2 x - 24) = 0$

$y = 2 x^{2} - 28$

Step 2.5.2

Factor.

Tap for more steps...

Step 2.5.2.1

Factor $x^{2} - 2 x - 24$ using the AC method.

Tap for more steps...

Step 2.5.2.1.1

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 $- 24$ and whose sum is $- 2$ .

$- 6, 4$

$y = 2 x^{2} - 28$

Step 2.5.2.1.2

Write the factored form using these integers.

$- 2 ((x - 6) (x + 4)) = 0$

$y = 2 x^{2} - 28$

$- 2 ((x - 6) (x + 4)) = 0$

$y = 2 x^{2} - 28$

Step 2.5.2.2

Remove unnecessary parentheses.

$- 2 (x - 6) (x + 4) = 0$

$y = 2 x^{2} - 28$

$- 2 (x - 6) (x + 4) = 0$

$y = 2 x^{2} - 28$

$- 2 (x - 6) (x + 4) = 0$

$y = 2 x^{2} - 28$

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 $0$ .

$x - 6 = 0$

$x + 4 = 0$

$y = 2 x^{2} - 28$

Step 2.7

Set $x - 6$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.7.1

Set $x - 6$ equal to $0$ .

$x - 6 = 0$

$y = 2 x^{2} - 28$

Step 2.7.2

Add $6$ to both sides of the equation.

$x = 6$

$y = 2 x^{2} - 28$

$x = 6$

$y = 2 x^{2} - 28$

Step 2.8

Set $x + 4$ equal to $0$ and solve for $x$ .

Tap for more steps...

Step 2.8.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

$y = 2 x^{2} - 28$

Step 2.8.2

Subtract $4$ from both sides of the equation.

$x = - 4$

$y = 2 x^{2} - 28$

$x = - 4$

$y = 2 x^{2} - 28$

Step 2.9

The final solution is all the values that make $- 2 (x - 6) (x + 4) = 0$ true.

$x = 6, - 4$

$y = 2 x^{2} - 28$

$x = 6, - 4$

$y = 2 x^{2} - 28$

Step 3

Replace all occurrences of $x$ with $6$ in each equation.

Tap for more steps...

Step 3.1

Replace all occurrences of $x$ in $y = 2 x^{2} - 28$ with $6$ .

$y = 2 {(6)}^{2} - 28$

$x = 6$

Step 3.2

Simplify the right side.

Tap for more steps...

Step 3.2.1

Simplify $2 {(6)}^{2} - 28$ .

Tap for more steps...

Step 3.2.1.1

Simplify each term.

Tap for more steps...

Step 3.2.1.1.1

Raise $6$ to the power of $2$ .

$y = 2 \cdot 36 - 28$

$x = 6$

Step 3.2.1.1.2

Multiply $2$ by $36$ .

$y = 72 - 28$

$x = 6$

$y = 72 - 28$

$x = 6$

Step 3.2.1.2

Subtract $28$ from $72$ .

$y = 44$

$x = 6$

$y = 44$

$x = 6$

$y = 44$

$x = 6$

$y = 44$

$x = 6$

Step 4

Replace all occurrences of $x$ with $- 4$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $x$ in $y = 2 x^{2} - 28$ with $- 4$ .

$y = 2 {(- 4)}^{2} - 28$

$x = - 4$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Simplify $2 {(- 4)}^{2} - 28$ .

Tap for more steps...

Step 4.2.1.1

Simplify each term.

Tap for more steps...

Step 4.2.1.1.1

Raise $- 4$ to the power of $2$ .

$y = 2 \cdot 16 - 28$

$x = - 4$

Step 4.2.1.1.2

Multiply $2$ by $16$ .

$y = 32 - 28$

$x = - 4$

$y = 32 - 28$

$x = - 4$

Step 4.2.1.2

Subtract $28$ from $32$ .

$y = 4$

$x = - 4$

$y = 4$

$x = - 4$

$y = 4$

$x = - 4$

$y = 4$

$x = - 4$

Step 5

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

$(6, 44)$

$(- 4, 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(6, 44), (- 4, 4)$

Equation Form:

$x = 6, y = 44$

$x = - 4, y = 4$

Step 7

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