Solve 75*8x-x^2=418*8

$$75 \cdot 8 x - x ^ { 2 } = 418 \cdot 8$$

Solve for x

$$x=8\sqrt{1354}+300\approx 594.373911888$$

$$x=300-8\sqrt{1354}\approx 5.626088112$$

Steps Using the Quadratic Formula

Multiply 75 and 8 to get 600.

$$600x-x^{2}=418\times 8$$

Multiply 418 and 8 to get 3344.

$$600x-x^{2}=3344$$

Subtract 3344 from both sides.

$$600x-x^{2}-3344=0$$

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.

$$-x^{2}+600x-3344=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 600 for b, and -3344 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-600±\sqrt{600^{2}-4\left(-1\right)\left(-3344\right)}}{2\left(-1\right)}$$

Square 600.

$$x=\frac{-600±\sqrt{360000-4\left(-1\right)\left(-3344\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-600±\sqrt{360000+4\left(-3344\right)}}{2\left(-1\right)}$$

Multiply 4 times -3344.

$$x=\frac{-600±\sqrt{360000-13376}}{2\left(-1\right)}$$

Add 360000 to -13376.

$$x=\frac{-600±\sqrt{346624}}{2\left(-1\right)}$$

Take the square root of 346624.

$$x=\frac{-600±16\sqrt{1354}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-600±16\sqrt{1354}}{-2}$$

Now solve the equation x=\frac{-600±16\sqrt{1354}}{-2} when ± is plus. Add -600 to 16\sqrt{1354}.

$$x=\frac{16\sqrt{1354}-600}{-2}$$

Divide -600+16\sqrt{1354} by -2.

$$x=300-8\sqrt{1354}$$

Now solve the equation x=\frac{-600±16\sqrt{1354}}{-2} when ± is minus. Subtract 16\sqrt{1354} from -600.

$$x=\frac{-16\sqrt{1354}-600}{-2}$$

Divide -600-16\sqrt{1354} by -2.

$$x=8\sqrt{1354}+300$$

The equation is now solved.

$$x=300-8\sqrt{1354} x=8\sqrt{1354}+300$$

Steps for Completing the Square

Multiply 75 and 8 to get 600.

$$600x-x^{2}=418\times 8$$

Multiply 418 and 8 to get 3344.

$$600x-x^{2}=3344$$

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.

$$-x^{2}+600x=3344$$

Divide both sides by -1.

$$\frac{-x^{2}+600x}{-1}=\frac{3344}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{600}{-1}x=\frac{3344}{-1}$$

Divide 600 by -1.

$$x^{2}-600x=\frac{3344}{-1}$$

Divide 3344 by -1.

$$x^{2}-600x=-3344$$

Divide -600, the coefficient of the x term, by 2 to get -300. Then add the square of -300 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-600x+\left(-300\right)^{2}=-3344+\left(-300\right)^{2}$$

Square -300.

$$x^{2}-600x+90000=-3344+90000$$

Add -3344 to 90000.

$$x^{2}-600x+90000=86656$$

Factor x^{2}-600x+90000. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-300\right)^{2}=86656$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-300\right)^{2}}=\sqrt{86656}$$

Simplify.

$$x-300=8\sqrt{1354} x-300=-8\sqrt{1354}$$

Add 300 to both sides of the equation.

$$x=8\sqrt{1354}+300 x=300-8\sqrt{1354}$$