Finite Math Examples

$x^{2} + 13 x + 36 = 0$ , $- x^{2} - 4 x + 2 = 0$ , $h = - 16 t^{2} + 32 t + 5$ , $2 x^{2} + 2 x - 5 = 0$

Step 1

Solve for $x$ in $x^{2} + 13 x + 36 = 0$ .

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Step 1.1

Factor $x^{2} + 13 x + 36$ using the AC method.

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Step 1.1.1

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 $36$ and whose sum is $13$ .

$4, 9$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 1.1.2

Write the factored form using these integers.

$(x + 4) (x + 9) = 0$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

$(x + 4) (x + 9) = 0$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 1.2

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

$x + 4 = 0$

$x + 9 = 0$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 1.3

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

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Step 1.3.1

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

$x + 4 = 0$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 1.3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

$x = - 4$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 1.4

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 1.4.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 1.4.2

Subtract $9$ from both sides of the equation.

$x = - 9$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

$x = - 9$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 1.5

The final solution is all the values that make $(x + 4) (x + 9) = 0$ true.

$x = - 4, - 9$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

$x = - 4, - 9$

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

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 2

Solve the system $x = - 4, - x^{2} - 4 x + 2 = 0, h = - 16 t^{2} + 32 t + 5, 2 x^{2} + 2 x - 5 = 0$ .

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Step 2.1

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

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Step 2.1.1

Replace all occurrences of $x$ in $- x^{2} - 4 x + 2 = 0$ with $- 4$ .

$- {(- 4)}^{2} - 4 \cdot - 4 + 2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 2.1.2

Simplify the left side.

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Step 2.1.2.1

Simplify $- {(- 4)}^{2} - 4 \cdot - 4 + 2$ .

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Step 2.1.2.1.1

Simplify each term.

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Step 2.1.2.1.1.1

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

$- 1 \cdot 16 - 4 \cdot - 4 + 2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 2.1.2.1.1.2

Multiply $- 1$ by $16$ .

$- 16 - 4 \cdot - 4 + 2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 2.1.2.1.1.3

Multiply $- 4$ by $- 4$ .

$- 16 + 16 + 2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

$- 16 + 16 + 2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 2.1.2.1.2

Simplify by adding numbers.

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Step 2.1.2.1.2.1

Add $- 16$ and $16$ .

$0 + 2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 2.1.2.1.2.2

Add $0$ and $2$ .

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$2 x^{2} + 2 x - 5 = 0$

Step 2.1.3

Replace all occurrences of $x$ in $2 x^{2} + 2 x - 5 = 0$ with $- 4$ .

$2 {(- 4)}^{2} + 2 (- 4) - 5 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

Step 2.1.4

Simplify the left side.

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Step 2.1.4.1

Simplify $2 {(- 4)}^{2} + 2 (- 4) - 5$ .

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Step 2.1.4.1.1

Simplify each term.

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Step 2.1.4.1.1.1

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

$2 \cdot 16 + 2 (- 4) - 5 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

Step 2.1.4.1.1.2

Multiply $2$ by $16$ .

$32 + 2 (- 4) - 5 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

Step 2.1.4.1.1.3

Multiply $2$ by $- 4$ .

$32 - 8 - 5 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$32 - 8 - 5 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

Step 2.1.4.1.2

Simplify by subtracting numbers.

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Step 2.1.4.1.2.1

Subtract $8$ from $32$ .

$24 - 5 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

Step 2.1.4.1.2.2

Subtract $5$ from $24$ .

$19 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$19 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$19 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$19 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

$19 = 0$

$2 = 0$

$x = - 4$

$h = - 16 t^{2} + 32 t + 5$

Step 2.2

Since $19 = 0$ is not true, there is no solution.

No solution