Solve by Factoring 2(a-5)(a+5)=21a

$2 (a - 5) (a + 5) = 21 a$

Step 1

Subtract $21 a$ from both sides of the equation.

$2 (a - 5) (a + 5) - 21 a = 0$

Step 2

Simplify each term.

Apply the distributive property.

$(2 a + 2 \cdot - 5) (a + 5) - 21 a = 0$

Multiply $2$ by $- 5$ .

$(2 a - 10) (a + 5) - 21 a = 0$

Expand $(2 a - 10) (a + 5)$ using the FOIL Method.

Apply the distributive property.

$2 a (a + 5) - 10 (a + 5) - 21 a = 0$

Apply the distributive property.

$2 a \cdot a + 2 a \cdot 5 - 10 (a + 5) - 21 a = 0$

Apply the distributive property.

$2 a \cdot a + 2 a \cdot 5 - 10 a - 10 \cdot 5 - 21 a = 0$

Simplify and combine like terms.

Simplify each term.

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

Move $a$ .

$2 (a \cdot a) + 2 a \cdot 5 - 10 a - 10 \cdot 5 - 21 a = 0$

Multiply $a$ by $a$ .

$2 a^{2} + 2 a \cdot 5 - 10 a - 10 \cdot 5 - 21 a = 0$

Multiply $5$ by $2$ .

$2 a^{2} + 10 a - 10 a - 10 \cdot 5 - 21 a = 0$

Multiply $- 10$ by $5$ .

$2 a^{2} + 10 a - 10 a - 50 - 21 a = 0$

Subtract $10 a$ from $10 a$ .

$2 a^{2} + 0 - 50 - 21 a = 0$

Add $2 a^{2}$ and $0$ .

$2 a^{2} - 50 - 21 a = 0$

Step 3

Factor by grouping.

Reorder terms.

$2 a^{2} - 21 a - 50 = 0$

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 50 = - 100$ and whose sum is $b = - 21$ .

Factor $- 21$ out of $- 21 a$ .

$2 a^{2} - 21 a - 50 = 0$

Rewrite $- 21$ as $4$ plus $- 25$

$2 a^{2} + (4 - 25) a - 50 = 0$

Apply the distributive property.

$2 a^{2} + 4 a - 25 a - 50 = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(2 a^{2} + 4 a) - 25 a - 50 = 0$

Factor out the greatest common factor (GCF) from each group.

$2 a (a + 2) - 25 (a + 2) = 0$

Factor the polynomial by factoring out the greatest common factor, $a + 2$ .

$(a + 2) (2 a - 25) = 0$

Step 4

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$a + 2 = 0$

$2 a - 25 = 0$

Step 5

Set $a + 2$ equal to $0$ and solve for $a$ .

Set $a + 2$ equal to $0$ .

$a + 2 = 0$

Subtract $2$ from both sides of the equation.

$a = - 2$

Step 6

Set $2 a - 25$ equal to $0$ and solve for $a$ .

Set $2 a - 25$ equal to $0$ .

$2 a - 25 = 0$

Solve $2 a - 25 = 0$ for $a$ .

Add $25$ to both sides of the equation.

$2 a = 25$

Divide each term in $2 a = 25$ by $2$ and simplify.

Divide each term in $2 a = 25$ by $2$ .

$\frac{2 a}{2} = \frac{25}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 a}{2} = \frac{25}{2}$

Divide $a$ by $1$ .

$a = \frac{25}{2}$

Step 7

The final solution is all the values that make $(a + 2) (2 a - 25) = 0$ true.

$a = - 2, \frac{25}{2}$

Solve by Factoring 2(a-5)(a+5)=21a

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