Calculus Examples

$64 x^{2} + 49 y^{2} + 896 x - 784 y + 3136 = 0$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$64 x^{2} + 49 {(0)}^{2} + 896 x - 784 \cdot 0 + 3136 = 0$

Step 1.2

Solve the equation.

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Step 1.2.1

Simplify $64 x^{2} + 49 {(0)}^{2} + 896 x - 784 \cdot 0 + 3136$ .

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Step 1.2.1.1

Simplify each term.

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Step 1.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$64 x^{2} + 49 \cdot 0 + 896 x - 784 \cdot 0 + 3136 = 0$

Step 1.2.1.1.2

Multiply $49$ by $0$ .

$64 x^{2} + 0 + 896 x - 784 \cdot 0 + 3136 = 0$

Step 1.2.1.1.3

Multiply $- 784$ by $0$ .

$64 x^{2} + 0 + 896 x + 0 + 3136 = 0$

Step 1.2.1.2

Combine the opposite terms in $64 x^{2} + 0 + 896 x + 0 + 3136$ .

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Step 1.2.1.2.1

Add $64 x^{2}$ and $0$ .

$64 x^{2} + 896 x + 0 + 3136 = 0$

Step 1.2.1.2.2

Add $64 x^{2} + 896 x$ and $0$ .

$64 x^{2} + 896 x + 3136 = 0$

Step 1.2.2

Factor the left side of the equation.

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Step 1.2.2.1

Factor $64$ out of $64 x^{2} + 896 x + 3136$ .

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Step 1.2.2.1.1

Factor $64$ out of $64 x^{2}$ .

$64 (x^{2}) + 896 x + 3136 = 0$

Step 1.2.2.1.2

Factor $64$ out of $896 x$ .

$64 (x^{2}) + 64 (14 x) + 3136 = 0$

Step 1.2.2.1.3

Factor $64$ out of $3136$ .

$64 x^{2} + 64 (14 x) + 64 \cdot 49 = 0$

Step 1.2.2.1.4

Factor $64$ out of $64 x^{2} + 64 (14 x)$ .

$64 (x^{2} + 14 x) + 64 \cdot 49 = 0$

Step 1.2.2.1.5

Factor $64$ out of $64 (x^{2} + 14 x) + 64 \cdot 49$ .

$64 (x^{2} + 14 x + 49) = 0$

Step 1.2.2.2

Factor using the perfect square rule.

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Step 1.2.2.2.1

Rewrite $49$ as $7^{2}$ .

$64 (x^{2} + 14 x + 7^{2}) = 0$

Step 1.2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$14 x = 2 \cdot x \cdot 7$

Step 1.2.2.2.3

Rewrite the polynomial.

$64 (x^{2} + 2 \cdot x \cdot 7 + 7^{2}) = 0$

Step 1.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 7$ .

$64 {(x + 7)}^{2} = 0$

Step 1.2.3

Divide each term in $64 {(x + 7)}^{2} = 0$ by $64$ and simplify.

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Step 1.2.3.1

Divide each term in $64 {(x + 7)}^{2} = 0$ by $64$ .

$\frac{64 {(x + 7)}^{2}}{64} = \frac{0}{64}$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Cancel the common factor of $64$ .

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Step 1.2.3.2.1.1

Cancel the common factor.

$\frac{64 {(x + 7)}^{2}}{64} = \frac{0}{64}$

Step 1.2.3.2.1.2

Divide ${(x + 7)}^{2}$ by $1$ .

${(x + 7)}^{2} = \frac{0}{64}$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Divide $0$ by $64$ .

${(x + 7)}^{2} = 0$

Step 1.2.4

Set the $x + 7$ equal to $0$ .

$x + 7 = 0$

Step 1.2.5

Subtract $7$ from both sides of the equation.

$x = - 7$

Step 1.3

x-intercept(s) in point form.

x-intercept(s): $(- 7, 0)$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$64 {(0)}^{2} + 49 y^{2} + 896 (0) - 784 y + 3136 = 0$

Step 2.2

Solve the equation.

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Step 2.2.1

Simplify $64 {(0)}^{2} + 49 y^{2} + 896 (0) - 784 y + 3136$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Raising $0$ to any positive power yields $0$ .

$64 \cdot 0 + 49 y^{2} + 896 (0) - 784 y + 3136 = 0$

Step 2.2.1.1.2

Multiply $64$ by $0$ .

$0 + 49 y^{2} + 896 (0) - 784 y + 3136 = 0$

Step 2.2.1.1.3

Multiply $896$ by $0$ .

$0 + 49 y^{2} + 0 - 784 y + 3136 = 0$

Step 2.2.1.2

Combine the opposite terms in $0 + 49 y^{2} + 0 - 784 y + 3136$ .

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Step 2.2.1.2.1

Add $0$ and $49 y^{2}$ .

$49 y^{2} + 0 - 784 y + 3136 = 0$

Step 2.2.1.2.2

Add $49 y^{2}$ and $0$ .

$49 y^{2} - 784 y + 3136 = 0$

Step 2.2.2

Factor the left side of the equation.

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Step 2.2.2.1

Factor $49$ out of $49 y^{2} - 784 y + 3136$ .

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Step 2.2.2.1.1

Factor $49$ out of $49 y^{2}$ .

$49 (y^{2}) - 784 y + 3136 = 0$

Step 2.2.2.1.2

Factor $49$ out of $- 784 y$ .

$49 (y^{2}) + 49 (- 16 y) + 3136 = 0$

Step 2.2.2.1.3

Factor $49$ out of $3136$ .

$49 y^{2} + 49 (- 16 y) + 49 \cdot 64 = 0$

Step 2.2.2.1.4

Factor $49$ out of $49 y^{2} + 49 (- 16 y)$ .

$49 (y^{2} - 16 y) + 49 \cdot 64 = 0$

Step 2.2.2.1.5

Factor $49$ out of $49 (y^{2} - 16 y) + 49 \cdot 64$ .

$49 (y^{2} - 16 y + 64) = 0$

Step 2.2.2.2

Factor using the perfect square rule.

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Step 2.2.2.2.1

Rewrite $64$ as $8^{2}$ .

$49 (y^{2} - 16 y + 8^{2}) = 0$

Step 2.2.2.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$16 y = 2 \cdot y \cdot 8$

Step 2.2.2.2.3

Rewrite the polynomial.

$49 (y^{2} - 2 \cdot y \cdot 8 + 8^{2}) = 0$

Step 2.2.2.2.4

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = y$ and $b = 8$ .

$49 {(y - 8)}^{2} = 0$

Step 2.2.3

Divide each term in $49 {(y - 8)}^{2} = 0$ by $49$ and simplify.

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Step 2.2.3.1

Divide each term in $49 {(y - 8)}^{2} = 0$ by $49$ .

$\frac{49 {(y - 8)}^{2}}{49} = \frac{0}{49}$

Step 2.2.3.2

Simplify the left side.

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Step 2.2.3.2.1

Cancel the common factor of $49$ .

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Step 2.2.3.2.1.1

Cancel the common factor.

$\frac{49 {(y - 8)}^{2}}{49} = \frac{0}{49}$

Step 2.2.3.2.1.2

Divide ${(y - 8)}^{2}$ by $1$ .

${(y - 8)}^{2} = \frac{0}{49}$

Step 2.2.3.3

Simplify the right side.

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Step 2.2.3.3.1

Divide $0$ by $49$ .

${(y - 8)}^{2} = 0$

Step 2.2.4

Set the $y - 8$ equal to $0$ .

$y - 8 = 0$

Step 2.2.5

Add $8$ to both sides of the equation.

$y = 8$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, 8)$

Step 3

List the intersections.

x-intercept(s): $(- 7, 0)$

y-intercept(s): $(0, 8)$

Step 4