Pre-Algebra Examples

$- 4 a (2 a - 7) = 33 - 8 a$

Step 1

Simplify $- 4 a (2 a - 7)$ .

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Step 1.1

Rewrite.

$0 + 0 - 4 a (2 a - 7) = 33 - 8 a$

Step 1.2

Simplify by multiplying through.

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Step 1.2.1

Apply the distributive property.

$- 4 a (2 a) - 4 a \cdot - 7 = 33 - 8 a$

Step 1.2.2

Simplify the expression.

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Step 1.2.2.1

Rewrite using the commutative property of multiplication.

$- 4 \cdot 2 a \cdot a - 4 a \cdot - 7 = 33 - 8 a$

Step 1.2.2.2

Multiply $- 7$ by $- 4$ .

$- 4 \cdot 2 a \cdot a + 28 a = 33 - 8 a$

Step 1.3

Simplify each term.

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Step 1.3.1

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

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Step 1.3.1.1

Move $a$ .

$- 4 \cdot 2 (a \cdot a) + 28 a = 33 - 8 a$

Step 1.3.1.2

Multiply $a$ by $a$ .

$- 4 \cdot 2 a^{2} + 28 a = 33 - 8 a$

Step 1.3.2

Multiply $- 4$ by $2$ .

$- 8 a^{2} + 28 a = 33 - 8 a$

Step 2

Move all terms containing $a$ to the left side of the equation.

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Step 2.1

Add $8 a$ to both sides of the equation.

$- 8 a^{2} + 28 a + 8 a = 33$

Step 2.2

Add $28 a$ and $8 a$ .

$- 8 a^{2} + 36 a = 33$

Step 3

Subtract $33$ from both sides of the equation.

$- 8 a^{2} + 36 a - 33 = 0$

Step 4

Use the quadratic formula to find the solutions.

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

Step 5

Substitute the values $a = - 8$ , $b = 36$ , and $c = - 33$ into the quadratic formula and solve for $a$ .

$\frac{- 36 \pm \sqrt{36^{2} - 4 \cdot (- 8 \cdot - 33)}}{2 \cdot - 8}$

Step 6

Simplify.

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Step 6.1

Simplify the numerator.

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Step 6.1.1

Raise $36$ to the power of $2$ .

$a = \frac{- 36 \pm \sqrt{1296 - 4 \cdot - 8 \cdot - 33}}{2 \cdot - 8}$

Step 6.1.2

Multiply $- 4 \cdot - 8 \cdot - 33$ .

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Step 6.1.2.1

Multiply $- 4$ by $- 8$ .

$a = \frac{- 36 \pm \sqrt{1296 + 32 \cdot - 33}}{2 \cdot - 8}$

Step 6.1.2.2

Multiply $32$ by $- 33$ .

$a = \frac{- 36 \pm \sqrt{1296 - 1056}}{2 \cdot - 8}$

Step 6.1.3

Subtract $1056$ from $1296$ .

$a = \frac{- 36 \pm \sqrt{240}}{2 \cdot - 8}$

Step 6.1.4

Rewrite $240$ as $4^{2} \cdot 15$ .

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Step 6.1.4.1

Factor $16$ out of $240$ .

$a = \frac{- 36 \pm \sqrt{16 (15)}}{2 \cdot - 8}$

Step 6.1.4.2

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

$a = \frac{- 36 \pm \sqrt{4^{2} \cdot 15}}{2 \cdot - 8}$

Step 6.1.5

Pull terms out from under the radical.

$a = \frac{- 36 \pm 4 \sqrt{15}}{2 \cdot - 8}$

Step 6.2

Multiply $2$ by $- 8$ .

$a = \frac{- 36 \pm 4 \sqrt{15}}{- 16}$

Step 6.3

Simplify $\frac{- 36 \pm 4 \sqrt{15}}{- 16}$ .

$a = \frac{9 \pm \sqrt{15}}{4}$

Step 7

The final answer is the combination of both solutions.

$a = \frac{9 + \sqrt{15}}{4}, \frac{9 - \sqrt{15}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$a = \frac{9 + \sqrt{15}}{4}, \frac{9 - \sqrt{15}}{4}$

Decimal Form:

$a = 3.21824583 \dots, 1.28175416 \dots$