Solve 38x=48(x+4)-2x(x-4)

$$38 x = 48 ( x + 4 ) - 2 x ( x - 4 )$$

Solve for x

$$x = \frac{\sqrt{465} + 9}{2} \approx 15.281929326$$

$$x=\frac{9-\sqrt{465}}{2}\approx -6.281929326$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 48 by x+4.

$$38x=48x+192-2x\left(x-4\right)$$

Add 2x\left(x-4\right) to both sides.

$$38x+2x\left(x-4\right)=48x+192$$

Use the distributive property to multiply 2x by x-4.

$$38x+2x^{2}-8x=48x+192$$

Combine 38x and -8x to get 30x.

$$30x+2x^{2}=48x+192$$

Subtract 48x from both sides.

$$30x+2x^{2}-48x=192$$

Combine 30x and -48x to get -18x.

$$-18x+2x^{2}=192$$

Subtract 192 from both sides.

$$-18x+2x^{2}-192=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$2x^{2}-18x-192=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -18 for b, and -192 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 2\left(-192\right)}}{2\times 2}$$

Square -18.

$$x=\frac{-\left(-18\right)±\sqrt{324-4\times 2\left(-192\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-18\right)±\sqrt{324-8\left(-192\right)}}{2\times 2}$$

Multiply -8 times -192.

$$x=\frac{-\left(-18\right)±\sqrt{324+1536}}{2\times 2}$$

Add 324 to 1536.

$$x=\frac{-\left(-18\right)±\sqrt{1860}}{2\times 2}$$

Take the square root of 1860.

$$x=\frac{-\left(-18\right)±2\sqrt{465}}{2\times 2}$$

The opposite of -18 is 18.

$$x=\frac{18±2\sqrt{465}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{18±2\sqrt{465}}{4}$$

Now solve the equation x=\frac{18±2\sqrt{465}}{4} when ± is plus. Add 18 to 2\sqrt{465}.

$$x=\frac{2\sqrt{465}+18}{4}$$

Divide 18+2\sqrt{465} by 4.

$$x=\frac{\sqrt{465}+9}{2}$$

Now solve the equation x=\frac{18±2\sqrt{465}}{4} when ± is minus. Subtract 2\sqrt{465} from 18.

$$x=\frac{18-2\sqrt{465}}{4}$$

Divide 18-2\sqrt{465} by 4.

$$x=\frac{9-\sqrt{465}}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{465}+9}{2} x=\frac{9-\sqrt{465}}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 48 by x+4.

$$38x=48x+192-2x\left(x-4\right)$$

Add 2x\left(x-4\right) to both sides.

$$38x+2x\left(x-4\right)=48x+192$$

Use the distributive property to multiply 2x by x-4.

$$38x+2x^{2}-8x=48x+192$$

Combine 38x and -8x to get 30x.

$$30x+2x^{2}=48x+192$$

Subtract 48x from both sides.

$$30x+2x^{2}-48x=192$$

Combine 30x and -48x to get -18x.

$$-18x+2x^{2}=192$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$2x^{2}-18x=192$$

Divide both sides by 2.

$$\frac{2x^{2}-18x}{2}=\frac{192}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{18}{2}\right)x=\frac{192}{2}$$

Divide -18 by 2.

$$x^{2}-9x=\frac{192}{2}$$

Divide 192 by 2.

$$x^{2}-9x=96$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=96+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=96+\frac{81}{4}$$

Add 96 to \frac{81}{4}.

$$x^{2}-9x+\frac{81}{4}=\frac{465}{4}$$

Factor x^{2}-9x+\frac{81}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{9}{2}\right)^{2}=\frac{465}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{\frac{465}{4}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{465}}{2} x-\frac{9}{2}=-\frac{\sqrt{465}}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{\sqrt{465}+9}{2} x=\frac{9-\sqrt{465}}{2}$$