Basic Math Examples

$(a + 1) (2 a - 3) = 30$

Step 1

Simplify $(a + 1) (2 a - 3)$ .

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Step 1.1

Expand $(a + 1) (2 a - 3)$ using the FOIL Method.

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Step 1.1.1

Apply the distributive property.

$a (2 a - 3) + 1 (2 a - 3) = 30$

Step 1.1.2

Apply the distributive property.

$a (2 a) + a \cdot - 3 + 1 (2 a - 3) = 30$

Step 1.1.3

Apply the distributive property.

$a (2 a) + a \cdot - 3 + 1 (2 a) + 1 \cdot - 3 = 30$

Step 1.2

Simplify and combine like terms.

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Step 1.2.1

Simplify each term.

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Step 1.2.1.1

Rewrite using the commutative property of multiplication.

$2 a \cdot a + a \cdot - 3 + 1 (2 a) + 1 \cdot - 3 = 30$

Step 1.2.1.2

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

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Step 1.2.1.2.1

Move $a$ .

$2 (a \cdot a) + a \cdot - 3 + 1 (2 a) + 1 \cdot - 3 = 30$

Step 1.2.1.2.2

Multiply $a$ by $a$ .

$2 a^{2} + a \cdot - 3 + 1 (2 a) + 1 \cdot - 3 = 30$

Step 1.2.1.3

Move $- 3$ to the left of $a$ .

$2 a^{2} - 3 \cdot a + 1 (2 a) + 1 \cdot - 3 = 30$

Step 1.2.1.4

Multiply $2 a$ by $1$ .

$2 a^{2} - 3 a + 2 a + 1 \cdot - 3 = 30$

Step 1.2.1.5

Multiply $- 3$ by $1$ .

$2 a^{2} - 3 a + 2 a - 3 = 30$

Step 1.2.2

Add $- 3 a$ and $2 a$ .

$2 a^{2} - a - 3 = 30$

Step 2

Move all terms to the left side of the equation and simplify.

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Step 2.1

Subtract $30$ from both sides of the equation.

$2 a^{2} - a - 3 - 30 = 0$

Step 2.2

Subtract $30$ from $- 3$ .

$2 a^{2} - a - 33 = 0$

Step 3

Use the quadratic formula to find the solutions.

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

Step 4

Substitute the values $a = 2$ , $b = - 1$ , and $c = - 33$ into the quadratic formula and solve for $a$ .

$\frac{1 \pm \sqrt{{(- 1)}^{2} - 4 \cdot (2 \cdot - 33)}}{2 \cdot 2}$

Step 5

Simplify.

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Step 5.1

Simplify the numerator.

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Step 5.1.1

Raise $- 1$ to the power of $2$ .

$a = \frac{1 \pm \sqrt{1 - 4 \cdot 2 \cdot - 33}}{2 \cdot 2}$

Step 5.1.2

Multiply $- 4 \cdot 2 \cdot - 33$ .

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Step 5.1.2.1

Multiply $- 4$ by $2$ .

$a = \frac{1 \pm \sqrt{1 - 8 \cdot - 33}}{2 \cdot 2}$

Step 5.1.2.2

Multiply $- 8$ by $- 33$ .

$a = \frac{1 \pm \sqrt{1 + 264}}{2 \cdot 2}$

Step 5.1.3

Add $1$ and $264$ .

$a = \frac{1 \pm \sqrt{265}}{2 \cdot 2}$

Step 5.2

Multiply $2$ by $2$ .

$a = \frac{1 \pm \sqrt{265}}{4}$

Step 6

The final answer is the combination of both solutions.

$a = \frac{1 + \sqrt{265}}{4}, \frac{1 - \sqrt{265}}{4}$

Step 7

The result can be shown in multiple forms.

Exact Form:

$a = \frac{1 + \sqrt{265}}{4}, \frac{1 - \sqrt{265}}{4}$

Decimal Form:

$a = 4.31970514 \dots, - 3.81970514 \dots$