Solve (40+2x)(24+2x)=1380

$$( 40 + 2 x ) ( 24 + 2 x ) = 1380$$

Solve for x

$$x=-35$$

$$x=3$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40+2x by 24+2x and combine like terms.

$$960+128x+4x^{2}=1380$$

Subtract 1380 from both sides.

$$960+128x+4x^{2}-1380=0$$

Subtract 1380 from 960 to get -420.

$$-420+128x+4x^{2}=0$$

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.

$$4x^{2}+128x-420=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 128 for b, and -420 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-128±\sqrt{128^{2}-4\times 4\left(-420\right)}}{2\times 4}$$

Square 128.

$$x=\frac{-128±\sqrt{16384-4\times 4\left(-420\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-128±\sqrt{16384-16\left(-420\right)}}{2\times 4}$$

Multiply -16 times -420.

$$x=\frac{-128±\sqrt{16384+6720}}{2\times 4}$$

Add 16384 to 6720.

$$x=\frac{-128±\sqrt{23104}}{2\times 4}$$

Take the square root of 23104.

$$x=\frac{-128±152}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-128±152}{8}$$

Now solve the equation x=\frac{-128±152}{8} when ± is plus. Add -128 to 152.

$$x=\frac{24}{8}$$

Divide 24 by 8.

$$x=3$$

Now solve the equation x=\frac{-128±152}{8} when ± is minus. Subtract 152 from -128.

$$x=-\frac{280}{8}$$

Divide -280 by 8.

$$x=-35$$

The equation is now solved.

$$x=3 x=-35$$

Steps for Completing the Square

Use the distributive property to multiply 40+2x by 24+2x and combine like terms.

$$960+128x+4x^{2}=1380$$

Subtract 960 from both sides.

$$128x+4x^{2}=1380-960$$

Subtract 960 from 1380 to get 420.

$$128x+4x^{2}=420$$

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.

$$4x^{2}+128x=420$$

Divide both sides by 4.

$$\frac{4x^{2}+128x}{4}=\frac{420}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{128}{4}x=\frac{420}{4}$$

Divide 128 by 4.

$$x^{2}+32x=\frac{420}{4}$$

Divide 420 by 4.

$$x^{2}+32x=105$$

Divide 32, the coefficient of the x term, by 2 to get 16. Then add the square of 16 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+32x+16^{2}=105+16^{2}$$

Square 16.

$$x^{2}+32x+256=105+256$$

Add 105 to 256.

$$x^{2}+32x+256=361$$

Factor x^{2}+32x+256. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+16\right)^{2}=361$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+16\right)^{2}}=\sqrt{361}$$

Simplify.

$$x+16=19 x+16=-19$$

Subtract 16 from both sides of the equation.

$$x=3 x=-35$$