Solve for y (4y+7)^2=4(4y+7)+6

${(4 y + 7)}^{2} = 4 (4 y + 7) + 6$

Step 1

Simplify $4 (4 y + 7) + 6$ .

Simplify each term.

Apply the distributive property.

${(4 y + 7)}^{2} = 4 (4 y) + 4 \cdot 7 + 6$

Multiply $4$ by $4$ .

${(4 y + 7)}^{2} = 16 y + 4 \cdot 7 + 6$

Multiply $4$ by $7$ .

${(4 y + 7)}^{2} = 16 y + 28 + 6$

Add $28$ and $6$ .

${(4 y + 7)}^{2} = 16 y + 34$

Step 2

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

Subtract $16 y$ from both sides of the equation.

${(4 y + 7)}^{2} - 16 y = 34$

Simplify each term.

Rewrite ${(4 y + 7)}^{2}$ as $(4 y + 7) (4 y + 7)$ .

$(4 y + 7) (4 y + 7) - 16 y = 34$

Expand $(4 y + 7) (4 y + 7)$ using the FOIL Method.

Apply the distributive property.

$4 y (4 y + 7) + 7 (4 y + 7) - 16 y = 34$

Apply the distributive property.

$4 y (4 y) + 4 y \cdot 7 + 7 (4 y + 7) - 16 y = 34$

Apply the distributive property.

$4 y (4 y) + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7 - 16 y = 34$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 \cdot 4 y \cdot y + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7 - 16 y = 34$

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

Move $y$ .

$4 \cdot 4 (y \cdot y) + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7 - 16 y = 34$

Multiply $y$ by $y$ .

$4 \cdot 4 y^{2} + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7 - 16 y = 34$

Multiply $4$ by $4$ .

$16 y^{2} + 4 y \cdot 7 + 7 (4 y) + 7 \cdot 7 - 16 y = 34$

Multiply $7$ by $4$ .

$16 y^{2} + 28 y + 7 (4 y) + 7 \cdot 7 - 16 y = 34$

Multiply $4$ by $7$ .

$16 y^{2} + 28 y + 28 y + 7 \cdot 7 - 16 y = 34$

Multiply $7$ by $7$ .

$16 y^{2} + 28 y + 28 y + 49 - 16 y = 34$

Add $28 y$ and $28 y$ .

$16 y^{2} + 56 y + 49 - 16 y = 34$

Subtract $16 y$ from $56 y$ .

$16 y^{2} + 40 y + 49 = 34$

Step 3

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

Subtract $34$ from both sides of the equation.

$16 y^{2} + 40 y + 49 - 34 = 0$

Subtract $34$ from $49$ .

$16 y^{2} + 40 y + 15 = 0$

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 $a = 16$ , $b = 40$ , and $c = 15$ into the quadratic formula and solve for $y$ .

$\frac{- 40 \pm \sqrt{40^{2} - 4 \cdot (16 \cdot 15)}}{2 \cdot 16}$

Step 6

Simplify.

Simplify the numerator.

Raise $40$ to the power of $2$ .

$y = \frac{- 40 \pm \sqrt{1600 - 4 \cdot 16 \cdot 15}}{2 \cdot 16}$

Multiply $- 4 \cdot 16 \cdot 15$ .

Multiply $- 4$ by $16$ .

$y = \frac{- 40 \pm \sqrt{1600 - 64 \cdot 15}}{2 \cdot 16}$

Multiply $- 64$ by $15$ .

$y = \frac{- 40 \pm \sqrt{1600 - 960}}{2 \cdot 16}$

Subtract $960$ from $1600$ .

$y = \frac{- 40 \pm \sqrt{640}}{2 \cdot 16}$

Rewrite $640$ as $8^{2} \cdot 10$ .

Factor $64$ out of $640$ .

$y = \frac{- 40 \pm \sqrt{64 (10)}}{2 \cdot 16}$

Rewrite $64$ as $8^{2}$ .

$y = \frac{- 40 \pm \sqrt{8^{2} \cdot 10}}{2 \cdot 16}$

Pull terms out from under the radical.

$y = \frac{- 40 \pm 8 \sqrt{10}}{2 \cdot 16}$

Multiply $2$ by $16$ .

$y = \frac{- 40 \pm 8 \sqrt{10}}{32}$

Simplify $\frac{- 40 \pm 8 \sqrt{10}}{32}$ .

$y = \frac{- 5 \pm \sqrt{10}}{4}$

Step 7

The final answer is the combination of both solutions.

$y = - \frac{5 - \sqrt{10}}{4}, - \frac{5 + \sqrt{10}}{4}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$y = - \frac{5 - \sqrt{10}}{4}, - \frac{5 + \sqrt{10}}{4}$

Decimal Form:

$y = - 0.45943058 \dots, - 2.04056941 \dots$

Solve for y (4y+7)^2=4(4y+7)+6

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