Basic Math Examples

$f (3 a) = 2 {(3 a)}^{2} - 3 (3 a) + 4$

Step 1

Since $a$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$2 {(3 a)}^{2} - 3 \cdot (3 a) + 4 = f \cdot (3 a)$

Step 2

Simplify each term.

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Step 2.1

Apply the product rule to $3 a$ .

$2 (3^{2} a^{2}) - 3 \cdot (3 a) + 4 = f \cdot (3 a)$

Step 2.2

Raise $3$ to the power of $2$ .

$2 (9 a^{2}) - 3 \cdot (3 a) + 4 = f \cdot (3 a)$

Step 2.3

Multiply $9$ by $2$ .

$18 a^{2} - 3 \cdot (3 a) + 4 = f \cdot (3 a)$

Step 2.4

Multiply $3$ by $- 3$ .

$18 a^{2} - 9 a + 4 = f \cdot (3 a)$

Step 3

Rewrite using the commutative property of multiplication.

$18 a^{2} - 9 a + 4 = 3 f a$

Step 4

Subtract $3 f a$ from both sides of the equation.

$18 a^{2} - 9 a + 4 - 3 f a = 0$

Step 5

Use the quadratic formula to find the solutions.

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

Step 6

Substitute the values $a = 18$ , $b = - 9 - 3 f$ , and $c = 4$ into the quadratic formula and solve for $a$ .

$\frac{- (- 9 - 3 f) \pm \sqrt{{(- 9 - 3 f)}^{2} - 4 \cdot (18 \cdot 4)}}{2 \cdot 18}$

Step 7

Simplify.

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Step 7.1

Simplify the numerator.

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Step 7.1.1

Apply the distributive property.

$a = \frac{9 - (- 3 f) \pm \sqrt{{(- 9 - 3 f)}^{2} - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.2

Multiply $- 1$ by $- 9$ .

$a = \frac{9 - (- 3 f) \pm \sqrt{{(- 9 - 3 f)}^{2} - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.3

Multiply $- 3$ by $- 1$ .

$a = \frac{9 + 3 f \pm \sqrt{{(- 9 - 3 f)}^{2} - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.4

Rewrite ${(- 9 - 3 f)}^{2}$ as $(- 9 - 3 f) (- 9 - 3 f)$ .

$a = \frac{9 + 3 f \pm \sqrt{(- 9 - 3 f) (- 9 - 3 f) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.5

Expand $(- 9 - 3 f) (- 9 - 3 f)$ using the FOIL Method.

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Step 7.1.5.1

Apply the distributive property.

$a = \frac{9 + 3 f \pm \sqrt{- 9 (- 9 - 3 f) - 3 f (- 9 - 3 f) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.5.2

Apply the distributive property.

$a = \frac{9 + 3 f \pm \sqrt{- 9 \cdot - 9 - 9 (- 3 f) - 3 f (- 9 - 3 f) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.5.3

Apply the distributive property.

$a = \frac{9 + 3 f \pm \sqrt{- 9 \cdot - 9 - 9 (- 3 f) - 3 f \cdot - 9 - 3 f (- 3 f) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.6

Simplify and combine like terms.

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Step 7.1.6.1

Simplify each term.

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Step 7.1.6.1.1

Multiply $- 9$ by $- 9$ .

$a = \frac{9 + 3 f \pm \sqrt{81 - 9 (- 3 f) - 3 f \cdot - 9 - 3 f (- 3 f) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.6.1.2

Multiply $- 3$ by $- 9$ .

$a = \frac{9 + 3 f \pm \sqrt{81 + 27 f - 3 f \cdot - 9 - 3 f (- 3 f) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.6.1.3

Multiply $- 9$ by $- 3$ .

$a = \frac{9 + 3 f \pm \sqrt{81 + 27 f + 27 f - 3 f (- 3 f) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.6.1.4

Rewrite using the commutative property of multiplication.

$a = \frac{9 + 3 f \pm \sqrt{81 + 27 f + 27 f - 3 \cdot (- 3 f \cdot f) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.6.1.5

Multiply $f$ by $f$ by adding the exponents.

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Step 7.1.6.1.5.1

Move $f$ .

$a = \frac{9 + 3 f \pm \sqrt{81 + 27 f + 27 f - 3 \cdot (- 3 (f \cdot f)) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.6.1.5.2

Multiply $f$ by $f$ .

$a = \frac{9 + 3 f \pm \sqrt{81 + 27 f + 27 f - 3 \cdot (- 3 f^{2}) - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.6.1.6

Multiply $- 3$ by $- 3$ .

$a = \frac{9 + 3 f \pm \sqrt{81 + 27 f + 27 f + 9 f^{2} - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.6.2

Add $27 f$ and $27 f$ .

$a = \frac{9 + 3 f \pm \sqrt{81 + 54 f + 9 f^{2} - 4 \cdot 18 \cdot 4}}{2 \cdot 18}$

Step 7.1.7

Multiply $- 4 \cdot 18 \cdot 4$ .

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Step 7.1.7.1

Multiply $- 4$ by $18$ .

$a = \frac{9 + 3 f \pm \sqrt{81 + 54 f + 9 f^{2} - 72 \cdot 4}}{2 \cdot 18}$

Step 7.1.7.2

Multiply $- 72$ by $4$ .

$a = \frac{9 + 3 f \pm \sqrt{81 + 54 f + 9 f^{2} - 288}}{2 \cdot 18}$

Step 7.1.8

Subtract $288$ from $81$ .

$a = \frac{9 + 3 f \pm \sqrt{54 f + 9 f^{2} - 207}}{2 \cdot 18}$

Step 7.1.9

Reorder terms.

$a = \frac{9 + 3 f \pm \sqrt{9 f^{2} + 54 f - 207}}{2 \cdot 18}$

Step 7.1.10

Factor $9$ out of $9 f^{2} + 54 f - 207$ .

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Step 7.1.10.1

Factor $9$ out of $9 f^{2}$ .

$a = \frac{9 + 3 f \pm \sqrt{9 (f^{2}) + 54 f - 207}}{2 \cdot 18}$

Step 7.1.10.2

Factor $9$ out of $54 f$ .

$a = \frac{9 + 3 f \pm \sqrt{9 (f^{2}) + 9 (6 f) - 207}}{2 \cdot 18}$

Step 7.1.10.3

Factor $9$ out of $- 207$ .

$a = \frac{9 + 3 f \pm \sqrt{9 f^{2} + 9 (6 f) + 9 \cdot - 23}}{2 \cdot 18}$

Step 7.1.10.4

Factor $9$ out of $9 f^{2} + 9 (6 f)$ .

$a = \frac{9 + 3 f \pm \sqrt{9 (f^{2} + 6 f) + 9 \cdot - 23}}{2 \cdot 18}$

Step 7.1.10.5

Factor $9$ out of $9 (f^{2} + 6 f) + 9 \cdot - 23$ .

$a = \frac{9 + 3 f \pm \sqrt{9 (f^{2} + 6 f - 23)}}{2 \cdot 18}$

Step 7.1.11

Rewrite $9$ as $3^{2}$ .

$a = \frac{9 + 3 f \pm \sqrt{3^{2} (f^{2} + 6 f - 23)}}{2 \cdot 18}$

Step 7.1.12

Pull terms out from under the radical.

$a = \frac{9 + 3 f \pm 3 \sqrt{f^{2} + 6 f - 23}}{2 \cdot 18}$

Step 7.2

Multiply $2$ by $18$ .

$a = \frac{9 + 3 f \pm 3 \sqrt{f^{2} + 6 f - 23}}{36}$

Step 8

The final answer is the combination of both solutions.

$a = \frac{3 + f + \sqrt{f^{2} + 6 f - 23}}{12}$

$a = \frac{3 + f - \sqrt{f^{2} + 6 f - 23}}{12}$