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

$$( 4 x + 2 ) ( x + 5 ) = x - 6$$

Solve for x

$$x=\frac{\sqrt{185}-21}{8}\approx -0.924816186$$

$$x=\frac{-\sqrt{185}-21}{8}\approx -4.325183814$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x+2 by x+5 and combine like terms.

$$4x^{2}+22x+10=x-6$$

Subtract x from both sides.

$$4x^{2}+22x+10-x=-6$$

Combine 22x and -x to get 21x.

$$4x^{2}+21x+10=-6$$

Add 6 to both sides.

$$4x^{2}+21x+10+6=0$$

Add 10 and 6 to get 16.

$$4x^{2}+21x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 21 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\times 4\times 16}}{2\times 4}$$

Square 21.

$$x=\frac{-21±\sqrt{441-4\times 4\times 16}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-21±\sqrt{441-16\times 16}}{2\times 4}$$

Multiply -16 times 16.

$$x=\frac{-21±\sqrt{441-256}}{2\times 4}$$

Add 441 to -256.

$$x=\frac{-21±\sqrt{185}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-21±\sqrt{185}}{8}$$

Now solve the equation x=\frac{-21±\sqrt{185}}{8} when ± is plus. Add -21 to \sqrt{185}.

$$x=\frac{\sqrt{185}-21}{8}$$

Now solve the equation x=\frac{-21±\sqrt{185}}{8} when ± is minus. Subtract \sqrt{185} from -21.

$$x=\frac{-\sqrt{185}-21}{8}$$

The equation is now solved.

$$x=\frac{\sqrt{185}-21}{8} x=\frac{-\sqrt{185}-21}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 4x+2 by x+5 and combine like terms.

$$4x^{2}+22x+10=x-6$$

Subtract x from both sides.

$$4x^{2}+22x+10-x=-6$$

Combine 22x and -x to get 21x.

$$4x^{2}+21x+10=-6$$

Subtract 10 from both sides.

$$4x^{2}+21x=-6-10$$

Subtract 10 from -6 to get -16.

$$4x^{2}+21x=-16$$

Divide both sides by 4.

$$\frac{4x^{2}+21x}{4}=-\frac{16}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{21}{4}x=-\frac{16}{4}$$

Divide -16 by 4.

$$x^{2}+\frac{21}{4}x=-4$$

Divide \frac{21}{4}, the coefficient of the x term, by 2 to get \frac{21}{8}. Then add the square of \frac{21}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{21}{4}x+\left(\frac{21}{8}\right)^{2}=-4+\left(\frac{21}{8}\right)^{2}$$

Square \frac{21}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{21}{4}x+\frac{441}{64}=-4+\frac{441}{64}$$

Add -4 to \frac{441}{64}.

$$x^{2}+\frac{21}{4}x+\frac{441}{64}=\frac{185}{64}$$

Factor x^{2}+\frac{21}{4}x+\frac{441}{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}.

$$\left(x+\frac{21}{8}\right)^{2}=\frac{185}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{21}{8}\right)^{2}}=\sqrt{\frac{185}{64}}$$

Simplify.

$$x+\frac{21}{8}=\frac{\sqrt{185}}{8} x+\frac{21}{8}=-\frac{\sqrt{185}}{8}$$

Subtract \frac{21}{8} from both sides of the equation.

$$x=\frac{\sqrt{185}-21}{8} x=\frac{-\sqrt{185}-21}{8}$$