Solve for a 49a^2=(5a-3y)^2

$49 a^{2} = {(5 a - 3 y)}^{2}$

Step 1

Simplify ${(5 a - 3 y)}^{2}$ .

Rewrite ${(5 a - 3 y)}^{2}$ as $(5 a - 3 y) (5 a - 3 y)$ .

$49 a^{2} = (5 a - 3 y) (5 a - 3 y)$

Expand $(5 a - 3 y) (5 a - 3 y)$ using the FOIL Method.

Apply the distributive property.

$49 a^{2} = 5 a (5 a - 3 y) - 3 y (5 a - 3 y)$

Apply the distributive property.

$49 a^{2} = 5 a (5 a) + 5 a (- 3 y) - 3 y (5 a - 3 y)$

Apply the distributive property.

$49 a^{2} = 5 a (5 a) + 5 a (- 3 y) - 3 y (5 a) - 3 y (- 3 y)$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$49 a^{2} = 5 \cdot 5 a \cdot a + 5 a (- 3 y) - 3 y (5 a) - 3 y (- 3 y)$

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

Move $a$ .

$49 a^{2} = 5 \cdot 5 (a \cdot a) + 5 a (- 3 y) - 3 y (5 a) - 3 y (- 3 y)$

Multiply $a$ by $a$ .

$49 a^{2} = 5 \cdot 5 a^{2} + 5 a (- 3 y) - 3 y (5 a) - 3 y (- 3 y)$

Multiply $5$ by $5$ .

$49 a^{2} = 25 a^{2} + 5 a (- 3 y) - 3 y (5 a) - 3 y (- 3 y)$

Rewrite using the commutative property of multiplication.

$49 a^{2} = 25 a^{2} + 5 \cdot - 3 a y - 3 y (5 a) - 3 y (- 3 y)$

Multiply $5$ by $- 3$ .

$49 a^{2} = 25 a^{2} - 15 a y - 3 y (5 a) - 3 y (- 3 y)$

Rewrite using the commutative property of multiplication.

$49 a^{2} = 25 a^{2} - 15 a y - 3 \cdot 5 y a - 3 y (- 3 y)$

Multiply $- 3$ by $5$ .

$49 a^{2} = 25 a^{2} - 15 a y - 15 y a - 3 y (- 3 y)$

Rewrite using the commutative property of multiplication.

$49 a^{2} = 25 a^{2} - 15 a y - 15 y a - 3 \cdot - 3 y \cdot y$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$49 a^{2} = 25 a^{2} - 15 a y - 15 y a - 3 \cdot - 3 (y \cdot y)$

Multiply $y$ by $y$ .

$49 a^{2} = 25 a^{2} - 15 a y - 15 y a - 3 \cdot - 3 y^{2}$

Multiply $- 3$ by $- 3$ .

$49 a^{2} = 25 a^{2} - 15 a y - 15 y a + 9 y^{2}$

Subtract $15 y a$ from $- 15 a y$ .

Move $y$ .

$49 a^{2} = 25 a^{2} - 15 a y - 15 a y + 9 y^{2}$

Subtract $15 a y$ from $- 15 a y$ .

$49 a^{2} = 25 a^{2} - 30 a y + 9 y^{2}$

Step 2

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

$25 a^{2} - 30 a y + 9 y^{2} = 49 a^{2}$

Step 3

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

Subtract $49 a^{2}$ from both sides of the equation.

$25 a^{2} - 30 a y + 9 y^{2} - 49 a^{2} = 0$

Subtract $49 a^{2}$ from $25 a^{2}$ .

$- 24 a^{2} - 30 a y + 9 y^{2} = 0$

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 $a = - 24$ , $b = - 30 y$ , and $c = 9 y^{2}$ into the quadratic formula and solve for $a$ .

$\frac{30 y \pm \sqrt{{(- 30 y)}^{2} - 4 \cdot (- 24 \cdot (9 y^{2}))}}{2 \cdot - 24}$

Step 6

Simplify.

Simplify the numerator.

Add parentheses.

$a = \frac{30 y \pm \sqrt{{(- 30 y)}^{2} - 4 \cdot (- 24 \cdot (9 y^{2}))}}{2 \cdot - 24}$

Let $u = - 24 \cdot (9 y^{2})$ . Substitute $u$ for all occurrences of $- 24 \cdot (9 y^{2})$ .

Apply the product rule to $- 30 y$ .

$a = \frac{30 y \pm \sqrt{{(- 30)}^{2} y^{2} - 4 \cdot u}}{2 \cdot - 24}$

Raise $- 30$ to the power of $2$ .

$a = \frac{30 y \pm \sqrt{900 y^{2} - 4 u}}{2 \cdot - 24}$

Factor $4$ out of $900 y^{2} - 4 u$ .

Factor $4$ out of $900 y^{2}$ .

$a = \frac{30 y \pm \sqrt{4 (225 y^{2}) - 4 u}}{2 \cdot - 24}$

Factor $4$ out of $- 4 u$ .

$a = \frac{30 y \pm \sqrt{4 (225 y^{2}) + 4 (- u)}}{2 \cdot - 24}$

Factor $4$ out of $4 (225 y^{2}) + 4 (- u)$ .

$a = \frac{30 y \pm \sqrt{4 (225 y^{2} - u)}}{2 \cdot - 24}$

Replace all occurrences of $u$ with $- 24 \cdot (9 y^{2})$ .

$a = \frac{30 y \pm \sqrt{4 (225 y^{2} - (- 24 \cdot (9 y^{2})))}}{2 \cdot - 24}$

Simplify.

Simplify each term.

Multiply $9$ by $- 24$ .

$a = \frac{30 y \pm \sqrt{4 (225 y^{2} - (- 216 y^{2}))}}{2 \cdot - 24}$

Multiply $- 216$ by $- 1$ .

$a = \frac{30 y \pm \sqrt{4 (225 y^{2} + 216 y^{2})}}{2 \cdot - 24}$

Add $225 y^{2}$ and $216 y^{2}$ .

$a = \frac{30 y \pm \sqrt{4 (441 y^{2})}}{2 \cdot - 24}$

$a = \frac{30 y \pm \sqrt{4 \cdot (441 y^{2})}}{2 \cdot - 24}$

Multiply $4$ by $441$ .

$a = \frac{30 y \pm \sqrt{1764 y^{2}}}{2 \cdot - 24}$

Rewrite $1764 y^{2}$ as ${(42 y)}^{2}$ .

$a = \frac{30 y \pm \sqrt{{(42 y)}^{2}}}{2 \cdot - 24}$

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

$a = \frac{30 y \pm 42 y}{2 \cdot - 24}$

Multiply $2$ by $- 24$ .

$a = \frac{30 y \pm 42 y}{- 48}$

Simplify $\frac{30 y \pm 42 y}{- 48}$ .

$a = \frac{5 y \pm 7 y}{- 8}$

Move the negative in front of the fraction.

$a = - \frac{5 y \pm 7 y}{8}$

Step 7

The final answer is the combination of both solutions.

$a = - \frac{3 y}{2}$

$a = \frac{y}{4}$

Solve for a 49a^2=(5a-3y)^2

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