Precalculus Examples

$y = a {(x - h)}^{2} + k \cdot 4 x^{2} + 8 x - 4$

Step 1

Rewrite the equation as $a {(x - h)}^{2} + k \cdot (4 x^{2}) + 8 x - 4 = y$ .

$a {(x - h)}^{2} + k \cdot (4 x^{2}) + 8 x - 4 = y$

Step 2

Rewrite using the commutative property of multiplication.

$a {(x - h)}^{2} + 4 k x^{2} + 8 x - 4 = y$

Step 3

Add $4$ to both sides of the equation.

$a {(x - h)}^{2} + 4 k x^{2} + 8 x = y + 4$

Step 4

Rewrite ${(x - h)}^{2}$ as $(x - h) (x - h)$ .

$a ((x - h) (x - h)) + 4 k x^{2} + 8 x = y + 4$

Step 5

Expand $(x - h) (x - h)$ using the FOIL Method.

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Step 5.1

Apply the distributive property.

$a (x (x - h) - h (x - h)) + 4 k x^{2} + 8 x = y + 4$

Step 5.2

Apply the distributive property.

$a (x \cdot x + x (- h) - h (x - h)) + 4 k x^{2} + 8 x = y + 4$

Step 5.3

Apply the distributive property.

$a (x \cdot x + x (- h) - h x - h (- h)) + 4 k x^{2} + 8 x = y + 4$

Step 6

Simplify and combine like terms.

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Step 6.1

Simplify each term.

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Step 6.1.1

Multiply $x$ by $x$ .

$a (x^{2} + x (- h) - h x - h (- h)) + 4 k x^{2} + 8 x = y + 4$

Step 6.1.2

Rewrite using the commutative property of multiplication.

$a (x^{2} - x h - h x - h (- h)) + 4 k x^{2} + 8 x = y + 4$

Step 6.1.3

Rewrite using the commutative property of multiplication.

$a (x^{2} - x h - h x - 1 \cdot (- 1 h \cdot h)) + 4 k x^{2} + 8 x = y + 4$

Step 6.1.4

Multiply $h$ by $h$ by adding the exponents.

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Step 6.1.4.1

Move $h$ .

$a (x^{2} - x h - h x - 1 \cdot (- 1 (h \cdot h))) + 4 k x^{2} + 8 x = y + 4$

Step 6.1.4.2

Multiply $h$ by $h$ .

$a (x^{2} - x h - h x - 1 \cdot (- 1 h^{2})) + 4 k x^{2} + 8 x = y + 4$

Step 6.1.5

Multiply $- 1$ by $- 1$ .

$a (x^{2} - x h - h x + 1 h^{2}) + 4 k x^{2} + 8 x = y + 4$

Step 6.1.6

Multiply $h^{2}$ by $1$ .

$a (x^{2} - x h - h x + h^{2}) + 4 k x^{2} + 8 x = y + 4$

Step 6.2

Subtract $h x$ from $- x h$ .

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Step 6.2.1

Move $x$ .

$a (x^{2} - h x - h x + h^{2}) + 4 k x^{2} + 8 x = y + 4$

Step 6.2.2

Subtract $h x$ from $- h x$ .

$a (x^{2} - 2 h x + h^{2}) + 4 k x^{2} + 8 x = y + 4$

Step 7

Apply the distributive property.

$a x^{2} + a (- 2 h x) + a h^{2} + 4 k x^{2} + 8 x = y + 4$

Step 8

Rewrite using the commutative property of multiplication.

$a x^{2} - 2 a h x + a h^{2} + 4 k x^{2} + 8 x = y + 4$

Step 9

Move all the expressions to the left side of the equation.

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Step 9.1

Subtract $y$ from both sides of the equation.

$a x^{2} - 2 a h x + a h^{2} + 4 k x^{2} + 8 x - y = 4$

Step 9.2

Subtract $4$ from both sides of the equation.

$a x^{2} - 2 a h x + a h^{2} + 4 k x^{2} + 8 x - y - 4 = 0$

Step 10

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 11

Substitute the values $a = a + 4 k$ , $b = - 2 a h + 8$ , and $c = a h^{2} - y - 4$ into the quadratic formula and solve for $x$ .

$\frac{- (- 2 a h + 8) \pm \sqrt{{(- 2 a h + 8)}^{2} - 4 \cdot ((a + 4 k) \cdot (a h^{2} - y - 4))}}{2 (a + 4 k)}$

Step 12

Simplify the numerator.

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Step 12.1

Apply the distributive property.

$x = \frac{- (- 2 a h) - 1 \cdot 8 \pm \sqrt{{(- 2 a h + 8)}^{2} - 4 \cdot (a + 4 k) \cdot (a h^{2} - y - 4)}}{2 (a + 4 k)}$

Step 12.2

Multiply $- 2$ by $- 1$ .

$x = \frac{2 (a h) - 1 \cdot 8 \pm \sqrt{{(- 2 a h + 8)}^{2} - 4 \cdot (a + 4 k) \cdot (a h^{2} - y - 4)}}{2 (a + 4 k)}$

Step 12.3

Multiply $- 1$ by $8$ .

$x = \frac{2 (a h) - 8 \pm \sqrt{{(- 2 a h + 8)}^{2} - 4 \cdot (a + 4 k) \cdot (a h^{2} - y - 4)}}{2 (a + 4 k)}$

Step 12.4

Add parentheses.

$x = \frac{2 a h - 8 \pm \sqrt{{(- 2 (a h) + 8)}^{2} - 4 \cdot ((a + 4 k) \cdot (a h^{2} - y - 4))}}{2 (a + 4 k)}$

Step 12.5

Let $u = (a + 4 k) \cdot (a h^{2} - y - 4)$ . Substitute $u$ for all occurrences of $(a + 4 k) \cdot (a h^{2} - y - 4)$ .

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Step 12.5.1

Rewrite ${(- 2 a h + 8)}^{2}$ as $(- 2 a h + 8) (- 2 a h + 8)$ .

$x = \frac{2 a h - 8 \pm \sqrt{(- 2 a h + 8) (- 2 a h + 8) - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.2

Expand $(- 2 a h + 8) (- 2 a h + 8)$ using the FOIL Method.

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Step 12.5.2.1

Apply the distributive property.

$x = \frac{2 a h - 8 \pm \sqrt{- 2 a h (- 2 a h + 8) + 8 (- 2 a h + 8) - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.2.2

Apply the distributive property.

$x = \frac{2 a h - 8 \pm \sqrt{- 2 a h (- 2 a h) - 2 a h \cdot 8 + 8 (- 2 a h + 8) - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.2.3

Apply the distributive property.

$x = \frac{2 a h - 8 \pm \sqrt{- 2 a h (- 2 a h) - 2 a h \cdot 8 + 8 (- 2 a h) + 8 \cdot 8 - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.3

Simplify and combine like terms.

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Step 12.5.3.1

Simplify each term.

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Step 12.5.3.1.1

Multiply $a$ by $a$ by adding the exponents.

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Step 12.5.3.1.1.1

Move $a$ .

$x = \frac{2 a h - 8 \pm \sqrt{- 2 (a \cdot a) h (- 2 h) - 2 a h \cdot 8 + 8 (- 2 a h) + 8 \cdot 8 - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.3.1.1.2

Multiply $a$ by $a$ .

$x = \frac{2 a h - 8 \pm \sqrt{- 2 a^{2} h (- 2 h) - 2 a h \cdot 8 + 8 (- 2 a h) + 8 \cdot 8 - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.3.1.2

Multiply $h$ by $h$ by adding the exponents.

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Step 12.5.3.1.2.1

Move $h$ .

$x = \frac{2 a h - 8 \pm \sqrt{- 2 a^{2} (h \cdot h) \cdot - 2 - 2 a h \cdot 8 + 8 (- 2 a h) + 8 \cdot 8 - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.3.1.2.2

Multiply $h$ by $h$ .

$x = \frac{2 a h - 8 \pm \sqrt{- 2 a^{2} h^{2} \cdot - 2 - 2 a h \cdot 8 + 8 (- 2 a h) + 8 \cdot 8 - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.3.1.3

Multiply $- 2$ by $- 2$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 a^{2} h^{2} - 2 a h \cdot 8 + 8 (- 2 a h) + 8 \cdot 8 - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.3.1.4

Multiply $8$ by $- 2$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 a^{2} h^{2} - 16 a h + 8 (- 2 a h) + 8 \cdot 8 - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.3.1.5

Multiply $- 2$ by $8$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 a^{2} h^{2} - 16 a h - 16 (a h) + 8 \cdot 8 - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.3.1.6

Multiply $8$ by $8$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 a^{2} h^{2} - 16 a h - 16 a h + 64 - 4 \cdot u}}{2 (a + 4 k)}$

Step 12.5.3.2

Subtract $16 a h$ from $- 16 a h$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 a^{2} h^{2} - 32 a h + 64 - 4 \cdot u}}{2 (a + 4 k)}$

$x = \frac{2 a h - 8 \pm \sqrt{4 a^{2} h^{2} - 32 a h + 64 - 4 u}}{2 (a + 4 k)}$

Step 12.6

Factor $4$ out of $4 a^{2} h^{2} - 32 a h + 64 - 4 u$ .

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Step 12.6.1

Factor $4$ out of $4 a^{2} h^{2}$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2}) - 32 a h + 64 - 4 u}}{2 (a + 4 k)}$

Step 12.6.2

Factor $4$ out of $- 32 a h$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2}) + 4 (- 8 a h) + 64 - 4 u}}{2 (a + 4 k)}$

Step 12.6.3

Factor $4$ out of $64$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2}) + 4 (- 8 a h) + 4 (16) - 4 u}}{2 (a + 4 k)}$

Step 12.6.4

Factor $4$ out of $- 4 u$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2}) + 4 (- 8 a h) + 4 (16) + 4 (- u)}}{2 (a + 4 k)}$

Step 12.6.5

Factor $4$ out of $4 (a^{2} h^{2}) + 4 (- 8 a h)$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h) + 4 (16) + 4 (- u)}}{2 (a + 4 k)}$

Step 12.6.6

Factor $4$ out of $4 (a^{2} h^{2} - 8 a h) + 4 (16)$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16) + 4 (- u)}}{2 (a + 4 k)}$

Step 12.6.7

Factor $4$ out of $4 (a^{2} h^{2} - 8 a h + 16) + 4 (- u)$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - u)}}{2 (a + 4 k)}$

Step 12.7

Replace all occurrences of $u$ with $(a + 4 k) \cdot (a h^{2} - y - 4)$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - ((a + 4 k) \cdot (a h^{2} - y - 4)))}}{2 (a + 4 k)}$

Step 12.8

Simplify.

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Step 12.8.1

Simplify each term.

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Step 12.8.1.1

Expand $(a + 4 k) (a h^{2} - y - 4)$ by multiplying each term in the first expression by each term in the second expression.

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - (a (a h^{2}) + a (- y) + a \cdot - 4 + 4 k (a h^{2}) + 4 k (- y) + 4 k \cdot - 4))}}{2 (a + 4 k)}$

Step 12.8.1.2

Simplify each term.

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Step 12.8.1.2.1

Multiply $a$ by $a$ by adding the exponents.

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Step 12.8.1.2.1.1

Move $a$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - (a \cdot a h^{2} + a (- y) + a \cdot - 4 + 4 k (a h^{2}) + 4 k (- y) + 4 k \cdot - 4))}}{2 (a + 4 k)}$

Step 12.8.1.2.1.2

Multiply $a$ by $a$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - (a^{2} h^{2} + a (- y) + a \cdot - 4 + 4 k (a h^{2}) + 4 k (- y) + 4 k \cdot - 4))}}{2 (a + 4 k)}$

Step 12.8.1.2.2

Rewrite using the commutative property of multiplication.

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - (a^{2} h^{2} - a y + a \cdot - 4 + 4 k (a h^{2}) + 4 k (- y) + 4 k \cdot - 4))}}{2 (a + 4 k)}$

Step 12.8.1.2.3

Move $- 4$ to the left of $a$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - (a^{2} h^{2} - a y - 4 \cdot a + 4 k (a h^{2}) + 4 k (- y) + 4 k \cdot - 4))}}{2 (a + 4 k)}$

Step 12.8.1.2.4

Rewrite using the commutative property of multiplication.

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - (a^{2} h^{2} - a y - 4 a + 4 k a h^{2} + 4 \cdot (- 1 k y) + 4 k \cdot - 4))}}{2 (a + 4 k)}$

Step 12.8.1.2.5

Multiply $4$ by $- 1$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - (a^{2} h^{2} - a y - 4 a + 4 k a h^{2} - 4 k y + 4 k \cdot - 4))}}{2 (a + 4 k)}$

Step 12.8.1.2.6

Multiply $- 4$ by $4$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - (a^{2} h^{2} - a y - 4 a + 4 k a h^{2} - 4 k y - 16 k))}}{2 (a + 4 k)}$

Step 12.8.1.3

Apply the distributive property.

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - (a^{2} h^{2}) - (- a y) - (- 4 a) - (4 k a h^{2}) - (- 4 k y) - (- 16 k))}}{2 (a + 4 k)}$

Step 12.8.1.4

Simplify.

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Step 12.8.1.4.1

Multiply $- (- a y)$ .

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Step 12.8.1.4.1.1

Multiply $- 1$ by $- 1$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - a^{2} h^{2} + 1 (a y) - (- 4 a) - (4 k a h^{2}) - (- 4 k y) - (- 16 k))}}{2 (a + 4 k)}$

Step 12.8.1.4.1.2

Multiply $a$ by $1$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - a^{2} h^{2} + a y - (- 4 a) - (4 k a h^{2}) - (- 4 k y) - (- 16 k))}}{2 (a + 4 k)}$

Step 12.8.1.4.2

Multiply $- 4$ by $- 1$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - a^{2} h^{2} + a y + 4 a - (4 k a h^{2}) - (- 4 k y) - (- 16 k))}}{2 (a + 4 k)}$

Step 12.8.1.4.3

Multiply $4$ by $- 1$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - a^{2} h^{2} + a y + 4 a - 4 (k a h^{2}) - (- 4 k y) - (- 16 k))}}{2 (a + 4 k)}$

Step 12.8.1.4.4

Multiply $- 4$ by $- 1$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - a^{2} h^{2} + a y + 4 a - 4 (k a h^{2}) + 4 (k y) - (- 16 k))}}{2 (a + 4 k)}$

Step 12.8.1.4.5

Multiply $- 16$ by $- 1$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - a^{2} h^{2} + a y + 4 a - 4 (k a h^{2}) + 4 (k y) + 16 k)}}{2 (a + 4 k)}$

Step 12.8.1.5

Remove parentheses.

$x = \frac{2 a h - 8 \pm \sqrt{4 (a^{2} h^{2} - 8 a h + 16 - a^{2} h^{2} + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k)}}{2 (a + 4 k)}$

Step 12.8.2

Combine the opposite terms in $a^{2} h^{2} - 8 a h + 16 - a^{2} h^{2} + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k$ .

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Step 12.8.2.1

Subtract $a^{2} h^{2}$ from $a^{2} h^{2}$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (- 8 a h + 16 + 0 + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k)}}{2 (a + 4 k)}$

Step 12.8.2.2

Add $- 8 a h + 16$ and $0$ .

$x = \frac{2 a h - 8 \pm \sqrt{4 (- 8 a h + 16 + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k)}}{2 (a + 4 k)}$

Step 12.9

Rewrite $4 (- 8 a h + 16 + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k)$ as $2^{2} (- 8 a h + 4^{2} + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k)$ .

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Step 12.9.1

Rewrite $4$ as $2^{2}$ .

$x = \frac{2 a h - 8 \pm \sqrt{2^{2} (- 8 a h + 16 + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k)}}{2 (a + 4 k)}$

Step 12.9.2

Rewrite $16$ as $4^{2}$ .

$x = \frac{2 a h - 8 \pm \sqrt{2^{2} (- 8 a h + 4^{2} + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k)}}{2 (a + 4 k)}$

Step 12.10

Pull terms out from under the radical.

$x = \frac{2 a h - 8 \pm 2 \sqrt{- 8 a h + 4^{2} + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k}}{2 (a + 4 k)}$

Step 12.11

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

$x = \frac{2 a h - 8 \pm 2 \sqrt{- 8 a h + 16 + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k}}{2 (a + 4 k)}$

Step 13

The final answer is the combination of both solutions.

$x = \frac{a h - 4 + \sqrt{- 8 a h + 16 + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k}}{a + 4 k}$

$x = \frac{a h - 4 - \sqrt{- 8 a h + 16 + a y + 4 a - 4 k a h^{2} + 4 k y + 16 k}}{a + 4 k}$