Solve Using the Quadratic Formula (a-9)(2a+1)=0

$(a - 9) (2 a + 1) = 0$

Step 1

Simplify the left side.

Simplify $(a - 9) (2 a + 1)$ .

Expand $(a - 9) (2 a + 1)$ using the FOIL Method.

Apply the distributive property.

$a (2 a + 1) - 9 (2 a + 1) = 0$

Apply the distributive property.

$a (2 a) + a \cdot 1 - 9 (2 a + 1) = 0$

Apply the distributive property.

$a (2 a) + a \cdot 1 - 9 (2 a) - 9 \cdot 1 = 0$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 a \cdot a + a \cdot 1 - 9 (2 a) - 9 \cdot 1 = 0$

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

Move $a$ .

$2 (a \cdot a) + a \cdot 1 - 9 (2 a) - 9 \cdot 1 = 0$

Multiply $a$ by $a$ .

$2 a^{2} + a \cdot 1 - 9 (2 a) - 9 \cdot 1 = 0$

Multiply $a$ by $1$ .

$2 a^{2} + a - 9 (2 a) - 9 \cdot 1 = 0$

Multiply $2$ by $- 9$ .

$2 a^{2} + a - 18 a - 9 \cdot 1 = 0$

Multiply $- 9$ by $1$ .

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

Subtract $18 a$ from $a$ .

$2 a^{2} - 17 a - 9 = 0$

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

Substitute the values $a = 2$ , $b = - 17$ , and $c = - 9$ into the quadratic formula and solve for $a$ .

$\frac{17 \pm \sqrt{{(- 17)}^{2} - 4 \cdot (2 \cdot - 9)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 17$ to the power of $2$ .

$a = \frac{17 \pm \sqrt{289 - 4 \cdot 2 \cdot - 9}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot - 9$ .

Multiply $- 4$ by $2$ .

$a = \frac{17 \pm \sqrt{289 - 8 \cdot - 9}}{2 \cdot 2}$

Multiply $- 8$ by $- 9$ .

$a = \frac{17 \pm \sqrt{289 + 72}}{2 \cdot 2}$

Add $289$ and $72$ .

$a = \frac{17 \pm \sqrt{361}}{2 \cdot 2}$

Rewrite $361$ as $19^{2}$ .

$a = \frac{17 \pm \sqrt{19^{2}}}{2 \cdot 2}$

Pull terms out from under the radical, assuming positive real numbers.

$a = \frac{17 \pm 19}{2 \cdot 2}$

Multiply $2$ by $2$ .

$a = \frac{17 \pm 19}{4}$

Step 5

The final answer is the combination of both solutions.

$a = 9, - \frac{1}{2}$

Solve Using the Quadratic Formula (a-9)(2a+1)=0

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