Solve 4x(x+5)=6x-4x^2

$$4 x ( x + 5 ) = 6 x - 4 x ^ { 2 }$$

Solve for x

$$x = -\frac{7}{4} = -1\frac{3}{4} = -1.75$$

$$x=0$$

Steps Using Factoring

Use the distributive property to multiply 4x by x+5.

$$4x^{2}+20x=6x-4x^{2}$$

Subtract 6x from both sides.

$$4x^{2}+20x-6x=-4x^{2}$$

Combine 20x and -6x to get 14x.

$$4x^{2}+14x=-4x^{2}$$

Add 4x^{2} to both sides.

$$4x^{2}+14x+4x^{2}=0$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}+14x=0$$

Factor out x.

$$x\left(8x+14\right)=0$$

To find equation solutions, solve x=0 and 8x+14=0.

$$x=0 x=-\frac{7}{4}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by x+5.

$$4x^{2}+20x=6x-4x^{2}$$

Subtract 6x from both sides.

$$4x^{2}+20x-6x=-4x^{2}$$

Combine 20x and -6x to get 14x.

$$4x^{2}+14x=-4x^{2}$$

Add 4x^{2} to both sides.

$$4x^{2}+14x+4x^{2}=0$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}+14x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, 14 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}}}{2\times 8}$$

Take the square root of 14^{2}.

$$x=\frac{-14±14}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-14±14}{16}$$

Now solve the equation x=\frac{-14±14}{16} when ± is plus. Add -14 to 14.

$$x=\frac{0}{16}$$

Divide 0 by 16.

$$x=0$$

Now solve the equation x=\frac{-14±14}{16} when ± is minus. Subtract 14 from -14.

$$x=-\frac{28}{16}$$

Reduce the fraction \frac{-28}{16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{7}{4}$$

The equation is now solved.

$$x=0 x=-\frac{7}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 4x by x+5.

$$4x^{2}+20x=6x-4x^{2}$$

Subtract 6x from both sides.

$$4x^{2}+20x-6x=-4x^{2}$$

Combine 20x and -6x to get 14x.

$$4x^{2}+14x=-4x^{2}$$

Add 4x^{2} to both sides.

$$4x^{2}+14x+4x^{2}=0$$

Combine 4x^{2} and 4x^{2} to get 8x^{2}.

$$8x^{2}+14x=0$$

Divide both sides by 8.

$$\frac{8x^{2}+14x}{8}=\frac{0}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\frac{14}{8}x=\frac{0}{8}$$

Reduce the fraction \frac{14}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{7}{4}x=\frac{0}{8}$$

Divide 0 by 8.

$$x^{2}+\frac{7}{4}x=0$$

Divide \frac{7}{4}, the coefficient of the x term, by 2 to get \frac{7}{8}. Then add the square of \frac{7}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{7}{4}x+\left(\frac{7}{8}\right)^{2}=\left(\frac{7}{8}\right)^{2}$$

Square \frac{7}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{4}x+\frac{49}{64}=\frac{49}{64}$$

Factor x^{2}+\frac{7}{4}x+\frac{49}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{8}\right)^{2}=\frac{49}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Simplify.

$$x+\frac{7}{8}=\frac{7}{8} x+\frac{7}{8}=-\frac{7}{8}$$

Subtract \frac{7}{8} from both sides of the equation.

$$x=0 x=-\frac{7}{4}$$