Solve (40+x)*(3/2x+60)=6420

$$( 40 + x ) \times ( \frac { 3 } { 2 } x + 60 ) = 6420$$

Solve for x

$$x=2\sqrt{1070}-40\approx 25.421708935$$

$$x=-2\sqrt{1070}-40\approx -105.421708935$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 40+x by \frac{3}{2}x+60 and combine like terms.

$$120x+2400+\frac{3}{2}x^{2}=6420$$

Subtract 6420 from both sides.

$$120x+2400+\frac{3}{2}x^{2}-6420=0$$

Subtract 6420 from 2400 to get -4020.

$$120x-4020+\frac{3}{2}x^{2}=0$$

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.

$$\frac{3}{2}x^{2}+120x-4020=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{3}{2} for a, 120 for b, and -4020 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\times \frac{3}{2}\left(-4020\right)}}{2\times \frac{3}{2}}$$

Square 120.

$$x=\frac{-120±\sqrt{14400-4\times \frac{3}{2}\left(-4020\right)}}{2\times \frac{3}{2}}$$

Multiply -4 times \frac{3}{2}.

$$x=\frac{-120±\sqrt{14400-6\left(-4020\right)}}{2\times \frac{3}{2}}$$

Multiply -6 times -4020.

$$x=\frac{-120±\sqrt{14400+24120}}{2\times \frac{3}{2}}$$

Add 14400 to 24120.

$$x=\frac{-120±\sqrt{38520}}{2\times \frac{3}{2}}$$

Take the square root of 38520.

$$x=\frac{-120±6\sqrt{1070}}{2\times \frac{3}{2}}$$

Multiply 2 times \frac{3}{2}.

$$x=\frac{-120±6\sqrt{1070}}{3}$$

Now solve the equation x=\frac{-120±6\sqrt{1070}}{3} when ± is plus. Add -120 to 6\sqrt{1070}.

$$x=\frac{6\sqrt{1070}-120}{3}$$

Divide -120+6\sqrt{1070} by 3.

$$x=2\sqrt{1070}-40$$

Now solve the equation x=\frac{-120±6\sqrt{1070}}{3} when ± is minus. Subtract 6\sqrt{1070} from -120.

$$x=\frac{-6\sqrt{1070}-120}{3}$$

Divide -120-6\sqrt{1070} by 3.

$$x=-2\sqrt{1070}-40$$

The equation is now solved.

$$x=2\sqrt{1070}-40 x=-2\sqrt{1070}-40$$

Steps for Completing the Square

Use the distributive property to multiply 40+x by \frac{3}{2}x+60 and combine like terms.

$$120x+2400+\frac{3}{2}x^{2}=6420$$

Subtract 2400 from both sides.

$$120x+\frac{3}{2}x^{2}=6420-2400$$

Subtract 2400 from 6420 to get 4020.

$$120x+\frac{3}{2}x^{2}=4020$$

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.

$$\frac{3}{2}x^{2}+120x=4020$$

Divide both sides of the equation by \frac{3}{2}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{\frac{3}{2}x^{2}+120x}{\frac{3}{2}}=\frac{4020}{\frac{3}{2}}$$

Dividing by \frac{3}{2} undoes the multiplication by \frac{3}{2}.

$$x^{2}+\frac{120}{\frac{3}{2}}x=\frac{4020}{\frac{3}{2}}$$

Divide 120 by \frac{3}{2} by multiplying 120 by the reciprocal of \frac{3}{2}.

$$x^{2}+80x=\frac{4020}{\frac{3}{2}}$$

Divide 4020 by \frac{3}{2} by multiplying 4020 by the reciprocal of \frac{3}{2}.

$$x^{2}+80x=2680$$

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.

$$x^{2}+80x+40^{2}=2680+40^{2}$$

Square 40.

$$x^{2}+80x+1600=2680+1600$$

Add 2680 to 1600.

$$x^{2}+80x+1600=4280$$

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}.

$$\left(x+40\right)^{2}=4280$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+40\right)^{2}}=\sqrt{4280}$$

Simplify.

$$x+40=2\sqrt{1070} x+40=-2\sqrt{1070}$$

Subtract 40 from both sides of the equation.

$$x=2\sqrt{1070}-40 x=-2\sqrt{1070}-40$$