Solve (40+x)(80+2x/3)[phone]=642000

$$( 40 + x ) ( \frac { 80 + 2 x } { 3 } ) + 130 - 60000 = 642000$$

Solve for x (complex solution)

$$x=\sqrt{1052805}-40\approx 986.062863571$$

$$x=-\left(\sqrt{1052805}+40\right)\approx -1066.062863571$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$\left(40+x\right)\left(80+2x\right)[phone]=1926000$$

Apply the distributive property by multiplying each term of 40+x by each term of 80+2x.

$$3200+80x+80x+2x^{2}[phone]=1926000$$

Combine 80x and 80x to get 160x.

$$3200+160x+2x^{2}[phone]=1926000$$

Add 3200 and 390 to get 3590.

$$3590+160x+2x^{2}-180000=1926000$$

Subtract 180000 from 3590 to get -176410.

$$-176410+160x+2x^{2}=1926000$$

Subtract 1926000 from both sides.

$$-176410+160x+2x^{2}-1926000=0$$

Subtract 1926000 from -176410 to get -2102410.

$$-2102410+160x+2x^{2}=0$$

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.

$$2x^{2}+160x-2102410=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 160 for b, and -2102410 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-160±\sqrt{160^{2}-4\times 2\left(-2102410\right)}}{2\times 2}$$

Square 160.

$$x=\frac{-160±\sqrt{25600-4\times 2\left(-2102410\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-160±\sqrt{25600-8\left(-2102410\right)}}{2\times 2}$$

Multiply -8 times -2102410.

$$x=\frac{-160±\sqrt{25600+16819280}}{2\times 2}$$

Add 25600 to 16819280.

$$x=\frac{-160±\sqrt{16844880}}{2\times 2}$$

Take the square root of 16844880.

$$x=\frac{-160±4\sqrt{1052805}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-160±4\sqrt{1052805}}{4}$$

Now solve the equation x=\frac{-160±4\sqrt{1052805}}{4} when ± is plus. Add -160 to 4\sqrt{1052805}.

$$x=\frac{4\sqrt{1052805}-160}{4}$$

Divide -160+4\sqrt{1052805} by 4.

$$x=\sqrt{1052805}-40$$

Now solve the equation x=\frac{-160±4\sqrt{1052805}}{4} when ± is minus. Subtract 4\sqrt{1052805} from -160.

$$x=\frac{-4\sqrt{1052805}-160}{4}$$

Divide -160-4\sqrt{1052805} by 4.

$$x=-\sqrt{1052805}-40$$

The equation is now solved.

$$x=\sqrt{1052805}-40 x=-\sqrt{1052805}-40$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$\left(40+x\right)\left(80+2x\right)[phone]=1926000$$

Apply the distributive property by multiplying each term of 40+x by each term of 80+2x.

$$3200+80x+80x+2x^{2}[phone]=1926000$$

Combine 80x and 80x to get 160x.

$$3200+160x+2x^{2}[phone]=1926000$$

Add 3200 and 390 to get 3590.

$$3590+160x+2x^{2}-180000=1926000$$

Subtract 180000 from 3590 to get -176410.

$$-176410+160x+2x^{2}=1926000$$

Add 176410 to both sides.

$$160x+2x^{2}=1926000+176410$$

Add 1926000 and 176410 to get 2102410.

$$160x+2x^{2}=2102410$$

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.

$$2x^{2}+160x=2102410$$

Divide both sides by 2.

$$\frac{2x^{2}+160x}{2}=\frac{2102410}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{160}{2}x=\frac{2102410}{2}$$

Divide 160 by 2.

$$x^{2}+80x=\frac{2102410}{2}$$

Divide 2102410 by 2.

$$x^{2}+80x=1051205$$

Divide 80, the coefficient of the x term, by 2 to get 40. Then add the square of 40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+80x+40^{2}=1051205+40^{2}$$

Square 40.

$$x^{2}+80x+1600=1051205+1600$$

Add 1051205 to 1600.

$$x^{2}+80x+1600=1052805$$

Factor x^{2}+80x+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+40\right)^{2}=1052805$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+40\right)^{2}}=\sqrt{1052805}$$

Simplify.

$$x+40=\sqrt{1052805} x+40=-\sqrt{1052805}$$

Subtract 40 from both sides of the equation.

$$x=\sqrt{1052805}-40 x=-\sqrt{1052805}-40$$

Solve for x

$$x=\sqrt{1052805}-40\approx 986.062863571$$

$$x=-\sqrt{1052805}-40\approx -1066.062863571$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 3.

$$\left(40+x\right)\left(80+2x\right)[phone]=1926000$$

Apply the distributive property by multiplying each term of 40+x by each term of 80+2x.

$$3200+80x+80x+2x^{2}[phone]=1926000$$

Combine 80x and 80x to get 160x.

$$3200+160x+2x^{2}[phone]=1926000$$

Add 3200 and 390 to get 3590.

$$3590+160x+2x^{2}-180000=1926000$$

Subtract 180000 from 3590 to get -176410.

$$-176410+160x+2x^{2}=1926000$$

Subtract 1926000 from both sides.

$$-176410+160x+2x^{2}-1926000=0$$

Subtract 1926000 from -176410 to get -2102410.

$$-2102410+160x+2x^{2}=0$$

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.

$$2x^{2}+160x-2102410=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 160 for b, and -2102410 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-160±\sqrt{160^{2}-4\times 2\left(-2102410\right)}}{2\times 2}$$

Square 160.

$$x=\frac{-160±\sqrt{25600-4\times 2\left(-2102410\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-160±\sqrt{25600-8\left(-2102410\right)}}{2\times 2}$$

Multiply -8 times -2102410.

$$x=\frac{-160±\sqrt{25600+16819280}}{2\times 2}$$

Add 25600 to 16819280.

$$x=\frac{-160±\sqrt{16844880}}{2\times 2}$$

Take the square root of 16844880.

$$x=\frac{-160±4\sqrt{1052805}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-160±4\sqrt{1052805}}{4}$$

Now solve the equation x=\frac{-160±4\sqrt{1052805}}{4} when ± is plus. Add -160 to 4\sqrt{1052805}.

$$x=\frac{4\sqrt{1052805}-160}{4}$$

Divide -160+4\sqrt{1052805} by 4.

$$x=\sqrt{1052805}-40$$

Now solve the equation x=\frac{-160±4\sqrt{1052805}}{4} when ± is minus. Subtract 4\sqrt{1052805} from -160.

$$x=\frac{-4\sqrt{1052805}-160}{4}$$

Divide -160-4\sqrt{1052805} by 4.

$$x=-\sqrt{1052805}-40$$

The equation is now solved.

$$x=\sqrt{1052805}-40 x=-\sqrt{1052805}-40$$

Steps for Completing the Square

Multiply both sides of the equation by 3.

$$\left(40+x\right)\left(80+2x\right)[phone]=1926000$$

Apply the distributive property by multiplying each term of 40+x by each term of 80+2x.

$$3200+80x+80x+2x^{2}[phone]=1926000$$

Combine 80x and 80x to get 160x.

$$3200+160x+2x^{2}[phone]=1926000$$

Add 3200 and 390 to get 3590.

$$3590+160x+2x^{2}-180000=1926000$$

Subtract 180000 from 3590 to get -176410.

$$-176410+160x+2x^{2}=1926000$$

Add 176410 to both sides.

$$160x+2x^{2}=1926000+176410$$

Add 1926000 and 176410 to get 2102410.

$$160x+2x^{2}=2102410$$

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.

$$2x^{2}+160x=2102410$$

Divide both sides by 2.

$$\frac{2x^{2}+160x}{2}=\frac{2102410}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{160}{2}x=\frac{2102410}{2}$$

Divide 160 by 2.

$$x^{2}+80x=\frac{2102410}{2}$$

Divide 2102410 by 2.

$$x^{2}+80x=1051205$$

Divide 80, the coefficient of the x term, by 2 to get 40. Then add the square of 40 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+80x+40^{2}=1051205+40^{2}$$

Square 40.

$$x^{2}+80x+1600=1051205+1600$$

Add 1051205 to 1600.

$$x^{2}+80x+1600=1052805$$

Factor x^{2}+80x+1600. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+40\right)^{2}=1052805$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+40\right)^{2}}=\sqrt{1052805}$$

Simplify.

$$x+40=\sqrt{1052805} x+40=-\sqrt{1052805}$$

Subtract 40 from both sides of the equation.

$$x=\sqrt{1052805}-40 x=-\sqrt{1052805}-40$$