Solve for f 2f(5f-2)-10(f^2-3f+6)=-8f(f+4)+4(2f^2-7f)

$2 f (5 f - 2) - 10 (f^{2} - 3 f + 6) = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Step 1

Simplify $2 f (5 f - 2) - 10 (f^{2} - 3 f + 6)$ .

Simplify each term.

Apply the distributive property.

$2 f (5 f) + 2 f \cdot - 2 - 10 (f^{2} - 3 f + 6) = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Rewrite using the commutative property of multiplication.

$2 \cdot 5 f \cdot f + 2 f \cdot - 2 - 10 (f^{2} - 3 f + 6) = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Multiply $- 2$ by $2$ .

$2 \cdot 5 f \cdot f - 4 f - 10 (f^{2} - 3 f + 6) = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Simplify each term.

Multiply $f$ by $f$ by adding the exponents.

Move $f$ .

$2 \cdot 5 (f \cdot f) - 4 f - 10 (f^{2} - 3 f + 6) = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Multiply $f$ by $f$ .

$2 \cdot 5 f^{2} - 4 f - 10 (f^{2} - 3 f + 6) = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Multiply $2$ by $5$ .

$10 f^{2} - 4 f - 10 (f^{2} - 3 f + 6) = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Apply the distributive property.

$10 f^{2} - 4 f - 10 f^{2} - 10 (- 3 f) - 10 \cdot 6 = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Simplify.

Multiply $- 3$ by $- 10$ .

$10 f^{2} - 4 f - 10 f^{2} + 30 f - 10 \cdot 6 = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Multiply $- 10$ by $6$ .

$10 f^{2} - 4 f - 10 f^{2} + 30 f - 60 = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Simplify by adding terms.

Combine the opposite terms in $10 f^{2} - 4 f - 10 f^{2} + 30 f - 60$ .

Subtract $10 f^{2}$ from $10 f^{2}$ .

$- 4 f + 0 + 30 f - 60 = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Add $- 4 f$ and $0$ .

$- 4 f + 30 f - 60 = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Add $- 4 f$ and $30 f$ .

$26 f - 60 = - 8 f (f + 4) + 4 (2 f^{2} - 7 f)$

Step 2

Simplify $- 8 f (f + 4) + 4 (2 f^{2} - 7 f)$ .

Simplify each term.

Apply the distributive property.

$26 f - 60 = - 8 f \cdot f - 8 f \cdot 4 + 4 (2 f^{2} - 7 f)$

Multiply $f$ by $f$ by adding the exponents.

Move $f$ .

$26 f - 60 = - 8 (f \cdot f) - 8 f \cdot 4 + 4 (2 f^{2} - 7 f)$

Multiply $f$ by $f$ .

$26 f - 60 = - 8 f^{2} - 8 f \cdot 4 + 4 (2 f^{2} - 7 f)$

Multiply $4$ by $- 8$ .

$26 f - 60 = - 8 f^{2} - 32 f + 4 (2 f^{2} - 7 f)$

Apply the distributive property.

$26 f - 60 = - 8 f^{2} - 32 f + 4 (2 f^{2}) + 4 (- 7 f)$

Multiply $2$ by $4$ .

$26 f - 60 = - 8 f^{2} - 32 f + 8 f^{2} + 4 (- 7 f)$

Multiply $- 7$ by $4$ .

$26 f - 60 = - 8 f^{2} - 32 f + 8 f^{2} - 28 f$

Simplify by adding terms.

Combine the opposite terms in $- 8 f^{2} - 32 f + 8 f^{2} - 28 f$ .

Add $- 8 f^{2}$ and $8 f^{2}$ .

$26 f - 60 = - 32 f + 0 - 28 f$

Add $- 32 f$ and $0$ .

$26 f - 60 = - 32 f - 28 f$

Subtract $28 f$ from $- 32 f$ .

$26 f - 60 = - 60 f$

Step 3

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

Add $60 f$ to both sides of the equation.

$26 f - 60 + 60 f = 0$

Add $26 f$ and $60 f$ .

$86 f - 60 = 0$

Step 4

Add $60$ to both sides of the equation.

$86 f = 60$

Step 5

Divide each term in $86 f = 60$ by $86$ and simplify.

Divide each term in $86 f = 60$ by $86$ .

$\frac{86 f}{86} = \frac{60}{86}$

Simplify the left side.

Cancel the common factor of $86$ .

Cancel the common factor.

$\frac{86 f}{86} = \frac{60}{86}$

Divide $f$ by $1$ .

$f = \frac{60}{86}$

Simplify the right side.

Cancel the common factor of $60$ and $86$ .

Factor $2$ out of $60$ .

$f = \frac{2 (30)}{86}$

Cancel the common factors.

Factor $2$ out of $86$ .

$f = \frac{2 \cdot 30}{2 \cdot 43}$

Cancel the common factor.

$f = \frac{2 \cdot 30}{2 \cdot 43}$

Rewrite the expression.

$f = \frac{30}{43}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$f = \frac{30}{43}$

Decimal Form:

$f = 0.69767441 \dots$

Solve for f 2f(5f-2)-10(f^2-3f+6)=-8f(f+4)+4(2f^2-7f)

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