Solve 23520-70w-0.7w^2/w=168

$$\frac { 23520 - 70 w - 0.7 w ^ { 2 } } { w } = 168$$

Solve for w

$$w=-420$$

$$w=80$$

Steps Using the Quadratic Formula

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w.

$$23520-70w-0.7w^{2}=168w$$

Subtract 168w from both sides.

$$23520-70w-0.7w^{2}-168w=0$$

Combine -70w and -168w to get -238w.

$$23520-238w-0.7w^{2}=0$$

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.

$$-0.7w^{2}-238w+23520=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -0.7 for a, -238 for b, and 23520 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$w=\frac{-\left(-238\right)±\sqrt{\left(-238\right)^{2}-4\left(-0.7\right)\times 23520}}{2\left(-0.7\right)}$$

Square -238.

$$w=\frac{-\left(-238\right)±\sqrt{56644-4\left(-0.7\right)\times 23520}}{2\left(-0.7\right)}$$

Multiply -4 times -0.7.

$$w=\frac{-\left(-238\right)±\sqrt{56644+2.8\times 23520}}{2\left(-0.7\right)}$$

Multiply 2.8 times 23520.

$$w=\frac{-\left(-238\right)±\sqrt{56644+65856}}{2\left(-0.7\right)}$$

Add 56644 to 65856.

$$w=\frac{-\left(-238\right)±\sqrt{122500}}{2\left(-0.7\right)}$$

Take the square root of 122500.

$$w=\frac{-\left(-238\right)±350}{2\left(-0.7\right)}$$

The opposite of -238 is 238.

$$w=\frac{238±350}{2\left(-0.7\right)}$$

Multiply 2 times -0.7.

$$w=\frac{238±350}{-1.4}$$

Now solve the equation w=\frac{238±350}{-1.4} when ± is plus. Add 238 to 350.

$$w=\frac{588}{-1.4}$$

Divide 588 by -1.4 by multiplying 588 by the reciprocal of -1.4.

$$w=-420$$

Now solve the equation w=\frac{238±350}{-1.4} when ± is minus. Subtract 350 from 238.

$$w=-\frac{112}{-1.4}$$

Divide -112 by -1.4 by multiplying -112 by the reciprocal of -1.4.

$$w=80$$

The equation is now solved.

$$w=-420 w=80$$

Steps for Completing the Square

Variable w cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by w.

$$23520-70w-0.7w^{2}=168w$$

Subtract 168w from both sides.

$$23520-70w-0.7w^{2}-168w=0$$

Combine -70w and -168w to get -238w.

$$23520-238w-0.7w^{2}=0$$

Subtract 23520 from both sides. Anything subtracted from zero gives its negation.

$$-238w-0.7w^{2}=-23520$$

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.

$$-0.7w^{2}-238w=-23520$$

Divide both sides of the equation by -0.7, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-0.7w^{2}-238w}{-0.7}=-\frac{23520}{-0.7}$$

Dividing by -0.7 undoes the multiplication by -0.7.

$$w^{2}+\left(-\frac{238}{-0.7}\right)w=-\frac{23520}{-0.7}$$

Divide -238 by -0.7 by multiplying -238 by the reciprocal of -0.7.

$$w^{2}+340w=-\frac{23520}{-0.7}$$

Divide -23520 by -0.7 by multiplying -23520 by the reciprocal of -0.7.

$$w^{2}+340w=33600$$

Divide 340, the coefficient of the x term, by 2 to get 170. Then add the square of 170 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$w^{2}+340w+170^{2}=33600+170^{2}$$

Square 170.

$$w^{2}+340w+28900=33600+28900$$

Add 33600 to 28900.

$$w^{2}+340w+28900=62500$$

Factor w^{2}+340w+28900. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(w+170\right)^{2}=62500$$

Take the square root of both sides of the equation.

$$\sqrt{\left(w+170\right)^{2}}=\sqrt{62500}$$

Simplify.

$$w+170=250 w+170=-250$$

Subtract 170 from both sides of the equation.

$$w=80 w=-420$$