Solve 5x(7x+6)=3x(4-8div5x)

$$5x(7x+6)=3x(4-8 \div 5x)$$

Solve for x

$$x=-\frac{90}{199}\approx -0.452261307$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 5x by 7x+6.

$$35x^{2}+30x=3x\left(4-\frac{8}{5}x\right)$$

Use the distributive property to multiply 3x by 4-\frac{8}{5}x.

$$35x^{2}+30x=12x+3x\left(-\frac{8}{5}\right)x$$

Multiply x and x to get x^{2}.

$$35x^{2}+30x=12x+3x^{2}\left(-\frac{8}{5}\right)$$

Express 3\left(-\frac{8}{5}\right) as a single fraction.

$$35x^{2}+30x=12x+\frac{3\left(-8\right)}{5}x^{2}$$

Multiply 3 and -8 to get -24.

$$35x^{2}+30x=12x+\frac{-24}{5}x^{2}$$

Fraction \frac{-24}{5} can be rewritten as -\frac{24}{5} by extracting the negative sign.

$$35x^{2}+30x=12x-\frac{24}{5}x^{2}$$

Subtract 12x from both sides.

$$35x^{2}+30x-12x=-\frac{24}{5}x^{2}$$

Combine 30x and -12x to get 18x.

$$35x^{2}+18x=-\frac{24}{5}x^{2}$$

Add \frac{24}{5}x^{2} to both sides.

$$35x^{2}+18x+\frac{24}{5}x^{2}=0$$

Combine 35x^{2} and \frac{24}{5}x^{2} to get \frac{199}{5}x^{2}.

$$\frac{199}{5}x^{2}+18x=0$$

Factor out x.

$$x\left(\frac{199}{5}x+18\right)=0$$

To find equation solutions, solve x=0 and \frac{199x}{5}+18=0.

$$x=0 x=-\frac{90}{199}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5x by 7x+6.

$$35x^{2}+30x=3x\left(4-\frac{8}{5}x\right)$$

Use the distributive property to multiply 3x by 4-\frac{8}{5}x.

$$35x^{2}+30x=12x+3x\left(-\frac{8}{5}\right)x$$

Multiply x and x to get x^{2}.

$$35x^{2}+30x=12x+3x^{2}\left(-\frac{8}{5}\right)$$

Express 3\left(-\frac{8}{5}\right) as a single fraction.

$$35x^{2}+30x=12x+\frac{3\left(-8\right)}{5}x^{2}$$

Multiply 3 and -8 to get -24.

$$35x^{2}+30x=12x+\frac{-24}{5}x^{2}$$

Fraction \frac{-24}{5} can be rewritten as -\frac{24}{5} by extracting the negative sign.

$$35x^{2}+30x=12x-\frac{24}{5}x^{2}$$

Subtract 12x from both sides.

$$35x^{2}+30x-12x=-\frac{24}{5}x^{2}$$

Combine 30x and -12x to get 18x.

$$35x^{2}+18x=-\frac{24}{5}x^{2}$$

Add \frac{24}{5}x^{2} to both sides.

$$35x^{2}+18x+\frac{24}{5}x^{2}=0$$

Combine 35x^{2} and \frac{24}{5}x^{2} to get \frac{199}{5}x^{2}.

$$\frac{199}{5}x^{2}+18x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{199}{5} for a, 18 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-18±\sqrt{18^{2}}}{2\times \frac{199}{5}}$$

Take the square root of 18^{2}.

$$x=\frac{-18±18}{2\times \frac{199}{5}}$$

Multiply 2 times \frac{199}{5}.

$$x=\frac{-18±18}{\frac{398}{5}}$$

Now solve the equation x=\frac{-18±18}{\frac{398}{5}} when ± is plus. Add -18 to 18.

$$x=\frac{0}{\frac{398}{5}}$$

Divide 0 by \frac{398}{5} by multiplying 0 by the reciprocal of \frac{398}{5}.

$$x=0$$

Now solve the equation x=\frac{-18±18}{\frac{398}{5}} when ± is minus. Subtract 18 from -18.

$$x=-\frac{36}{\frac{398}{5}}$$

Divide -36 by \frac{398}{5} by multiplying -36 by the reciprocal of \frac{398}{5}.

$$x=-\frac{90}{199}$$

The equation is now solved.

$$x=0 x=-\frac{90}{199}$$

Steps for Completing the Square

Use the distributive property to multiply 5x by 7x+6.

$$35x^{2}+30x=3x\left(4-\frac{8}{5}x\right)$$

Use the distributive property to multiply 3x by 4-\frac{8}{5}x.

$$35x^{2}+30x=12x+3x\left(-\frac{8}{5}\right)x$$

Multiply x and x to get x^{2}.

$$35x^{2}+30x=12x+3x^{2}\left(-\frac{8}{5}\right)$$

Express 3\left(-\frac{8}{5}\right) as a single fraction.

$$35x^{2}+30x=12x+\frac{3\left(-8\right)}{5}x^{2}$$

Multiply 3 and -8 to get -24.

$$35x^{2}+30x=12x+\frac{-24}{5}x^{2}$$

Fraction \frac{-24}{5} can be rewritten as -\frac{24}{5} by extracting the negative sign.

$$35x^{2}+30x=12x-\frac{24}{5}x^{2}$$

Subtract 12x from both sides.

$$35x^{2}+30x-12x=-\frac{24}{5}x^{2}$$

Combine 30x and -12x to get 18x.

$$35x^{2}+18x=-\frac{24}{5}x^{2}$$

Add \frac{24}{5}x^{2} to both sides.

$$35x^{2}+18x+\frac{24}{5}x^{2}=0$$

Combine 35x^{2} and \frac{24}{5}x^{2} to get \frac{199}{5}x^{2}.

$$\frac{199}{5}x^{2}+18x=0$$

Divide both sides of the equation by \frac{199}{5}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{199}{5}x^{2}+18x}{\frac{199}{5}}=\frac{0}{\frac{199}{5}}$$

Dividing by \frac{199}{5} undoes the multiplication by \frac{199}{5}.

$$x^{2}+\frac{18}{\frac{199}{5}}x=\frac{0}{\frac{199}{5}}$$

Divide 18 by \frac{199}{5} by multiplying 18 by the reciprocal of \frac{199}{5}.

$$x^{2}+\frac{90}{199}x=\frac{0}{\frac{199}{5}}$$

Divide 0 by \frac{199}{5} by multiplying 0 by the reciprocal of \frac{199}{5}.

$$x^{2}+\frac{90}{199}x=0$$

Divide \frac{90}{199}, the coefficient of the x term, by 2 to get \frac{45}{199}. Then add the square of \frac{45}{199} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{90}{199}x+\left(\frac{45}{199}\right)^{2}=\left(\frac{45}{199}\right)^{2}$$

Square \frac{45}{199} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{90}{199}x+\frac{2025}{39601}=\frac{2025}{39601}$$

Factor x^{2}+\frac{90}{199}x+\frac{2025}{39601}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{45}{199}\right)^{2}=\frac{2025}{39601}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{45}{199}\right)^{2}}=\sqrt{\frac{2025}{39601}}$$

Simplify.

$$x+\frac{45}{199}=\frac{45}{199} x+\frac{45}{199}=-\frac{45}{199}$$

Subtract \frac{45}{199} from both sides of the equation.

$$x=0 x=-\frac{90}{199}$$