Solve 4x(x+3)+2(x+1)=74

$$4 x ( x + 3 ) + 2 ( x + 1 ) = 74$$

Solve for x

$$x = \frac{\sqrt{337} - 7}{4} \approx 2.839389938$$

$$x=\frac{-\sqrt{337}-7}{4}\approx -6.339389938$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 4x by x+3.

$$4x^{2}+12x+2\left(x+1\right)=74$$

Use the distributive property to multiply 2 by x+1.

$$4x^{2}+12x+2x+2=74$$

Combine 12x and 2x to get 14x.

$$4x^{2}+14x+2=74$$

Subtract 74 from both sides.

$$4x^{2}+14x+2-74=0$$

Subtract 74 from 2 to get -72.

$$4x^{2}+14x-72=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 14 for b, and -72 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 4\left(-72\right)}}{2\times 4}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 4\left(-72\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-14±\sqrt{196-16\left(-72\right)}}{2\times 4}$$

Multiply -16 times -72.

$$x=\frac{-14±\sqrt{196+1152}}{2\times 4}$$

Add 196 to 1152.

$$x=\frac{-14±\sqrt{1348}}{2\times 4}$$

Take the square root of 1348.

$$x=\frac{-14±2\sqrt{337}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-14±2\sqrt{337}}{8}$$

Now solve the equation x=\frac{-14±2\sqrt{337}}{8} when ± is plus. Add -14 to 2\sqrt{337}.

$$x=\frac{2\sqrt{337}-14}{8}$$

Divide -14+2\sqrt{337} by 8.

$$x=\frac{\sqrt{337}-7}{4}$$

Now solve the equation x=\frac{-14±2\sqrt{337}}{8} when ± is minus. Subtract 2\sqrt{337} from -14.

$$x=\frac{-2\sqrt{337}-14}{8}$$

Divide -14-2\sqrt{337} by 8.

$$x=\frac{-\sqrt{337}-7}{4}$$

The equation is now solved.

$$x=\frac{\sqrt{337}-7}{4} x=\frac{-\sqrt{337}-7}{4}$$

Steps for Completing the Square

Use the distributive property to multiply 4x by x+3.

$$4x^{2}+12x+2\left(x+1\right)=74$$

Use the distributive property to multiply 2 by x+1.

$$4x^{2}+12x+2x+2=74$$

Combine 12x and 2x to get 14x.

$$4x^{2}+14x+2=74$$

Subtract 2 from both sides.

$$4x^{2}+14x=74-2$$

Subtract 2 from 74 to get 72.

$$4x^{2}+14x=72$$

Divide both sides by 4.

$$\frac{4x^{2}+14x}{4}=\frac{72}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{14}{4}x=\frac{72}{4}$$

Reduce the fraction \frac{14}{4} to lowest terms by extracting and canceling out 2.

$$x^{2}+\frac{7}{2}x=\frac{72}{4}$$

Divide 72 by 4.

$$x^{2}+\frac{7}{2}x=18$$

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

$$x^{2}+\frac{7}{2}x+\left(\frac{7}{4}\right)^{2}=18+\left(\frac{7}{4}\right)^{2}$$

Square \frac{7}{4} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=18+\frac{49}{16}$$

Add 18 to \frac{49}{16}.

$$x^{2}+\frac{7}{2}x+\frac{49}{16}=\frac{337}{16}$$

Factor x^{2}+\frac{7}{2}x+\frac{49}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{7}{4}\right)^{2}=\frac{337}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{4}\right)^{2}}=\sqrt{\frac{337}{16}}$$

Simplify.

$$x+\frac{7}{4}=\frac{\sqrt{337}}{4} x+\frac{7}{4}=-\frac{\sqrt{337}}{4}$$

Subtract \frac{7}{4} from both sides of the equation.

$$x=\frac{\sqrt{337}-7}{4} x=\frac{-\sqrt{337}-7}{4}$$