Algebra Examples

$- {(8 x + 4)}^{2} - 3$

Step 1

Write $- {(8 x + 4)}^{2} - 3$ as an equation.

$y = - {(8 x + 4)}^{2} - 3$

Step 2

Find the x-intercepts.

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Step 2.1

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

$0 = - {(8 x + 4)}^{2} - 3$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$0 = - {(8 x + 4)}^{2} - 3$

Step 2.2.2

Simplify $- {(8 x + 4)}^{2} - 3$ .

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Step 2.2.2.1

Simplify each term.

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Step 2.2.2.1.1

Rewrite ${(8 x + 4)}^{2}$ as $(8 x + 4) (8 x + 4)$ .

$0 = - ((8 x + 4) (8 x + 4)) - 3$

Step 2.2.2.1.2

Expand $(8 x + 4) (8 x + 4)$ using the FOIL Method.

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Step 2.2.2.1.2.1

Apply the distributive property.

$0 = - (8 x (8 x + 4) + 4 (8 x + 4)) - 3$

Step 2.2.2.1.2.2

Apply the distributive property.

$0 = - (8 x (8 x) + 8 x \cdot 4 + 4 (8 x + 4)) - 3$

Step 2.2.2.1.2.3

Apply the distributive property.

$0 = - (8 x (8 x) + 8 x \cdot 4 + 4 (8 x) + 4 \cdot 4) - 3$

Step 2.2.2.1.3

Simplify and combine like terms.

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Step 2.2.2.1.3.1

Simplify each term.

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Step 2.2.2.1.3.1.1

Rewrite using the commutative property of multiplication.

$0 = - (8 \cdot 8 x \cdot x + 8 x \cdot 4 + 4 (8 x) + 4 \cdot 4) - 3$

Step 2.2.2.1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 2.2.2.1.3.1.2.1

Move $x$ .

$0 = - (8 \cdot 8 (x \cdot x) + 8 x \cdot 4 + 4 (8 x) + 4 \cdot 4) - 3$

Step 2.2.2.1.3.1.2.2

Multiply $x$ by $x$ .

$0 = - (8 \cdot 8 x^{2} + 8 x \cdot 4 + 4 (8 x) + 4 \cdot 4) - 3$

Step 2.2.2.1.3.1.3

Multiply $8$ by $8$ .

$0 = - (64 x^{2} + 8 x \cdot 4 + 4 (8 x) + 4 \cdot 4) - 3$

Step 2.2.2.1.3.1.4

Multiply $4$ by $8$ .

$0 = - (64 x^{2} + 32 x + 4 (8 x) + 4 \cdot 4) - 3$

Step 2.2.2.1.3.1.5

Multiply $8$ by $4$ .

$0 = - (64 x^{2} + 32 x + 32 x + 4 \cdot 4) - 3$

Step 2.2.2.1.3.1.6

Multiply $4$ by $4$ .

$0 = - (64 x^{2} + 32 x + 32 x + 16) - 3$

Step 2.2.2.1.3.2

Add $32 x$ and $32 x$ .

$0 = - (64 x^{2} + 64 x + 16) - 3$

Step 2.2.2.1.4

Apply the distributive property.

$0 = - (64 x^{2}) - (64 x) - 1 \cdot 16 - 3$

Step 2.2.2.1.5

Simplify.

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Step 2.2.2.1.5.1

Multiply $64$ by $- 1$ .

$0 = - 64 x^{2} - (64 x) - 1 \cdot 16 - 3$

Step 2.2.2.1.5.2

Multiply $64$ by $- 1$ .

$0 = - 64 x^{2} - 64 x - 1 \cdot 16 - 3$

Step 2.2.2.1.5.3

Multiply $- 1$ by $16$ .

$0 = - 64 x^{2} - 64 x - 16 - 3$

Step 2.2.2.2

Subtract $3$ from $- 16$ .

$0 = - 64 x^{2} - 64 x - 19$

Step 2.2.3

Graph each side of the equation. The solution is the x-value of the point of intersection.

No solution

Step 2.3

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

x-intercept(s): $None$

Step 3

Find the y-intercepts.

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Step 3.1

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

$y = - {(8 (0) + 4)}^{2} - 3$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = - {(8 (0) + 4)}^{2} - 3$

Step 3.2.2

Simplify $- {(8 (0) + 4)}^{2} - 3$ .

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Step 3.2.2.1

Simplify each term.

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Step 3.2.2.1.1

Multiply $8$ by $0$ .

$y = - {(0 + 4)}^{2} - 3$

Step 3.2.2.1.2

Add $0$ and $4$ .

$y = - 4^{2} - 3$

Step 3.2.2.1.3

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

$y = - 1 \cdot 16 - 3$

Step 3.2.2.1.4

Multiply $- 1$ by $16$ .

$y = - 16 - 3$

Step 3.2.2.2

Subtract $3$ from $- 16$ .

$y = - 19$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, - 19)$

Step 4

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, - 19)$

Step 5