Solve 300=30x+2x^2

$$300 = 30 x + 2 x ^ { 2 }$$

Solve for x

$$x = \frac{5 \sqrt{33} - 15}{2} \approx 6.861406616$$

$$x=\frac{-5\sqrt{33}-15}{2}\approx -21.861406616$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$30x+2x^{2}=300$$

Subtract 300 from both sides.

$$30x+2x^{2}-300=0$$

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.

$$2x^{2}+30x-300=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 30 for b, and -300 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 2\left(-300\right)}}{2\times 2}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 2\left(-300\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-30±\sqrt{900-8\left(-300\right)}}{2\times 2}$$

Multiply -8 times -300.

$$x=\frac{-30±\sqrt{900+2400}}{2\times 2}$$

Add 900 to 2400.

$$x=\frac{-30±\sqrt{3300}}{2\times 2}$$

Take the square root of 3300.

$$x=\frac{-30±10\sqrt{33}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-30±10\sqrt{33}}{4}$$

Now solve the equation x=\frac{-30±10\sqrt{33}}{4} when ± is plus. Add -30 to 10\sqrt{33}.

$$x=\frac{10\sqrt{33}-30}{4}$$

Divide -30+10\sqrt{33} by 4.

$$x=\frac{5\sqrt{33}-15}{2}$$

Now solve the equation x=\frac{-30±10\sqrt{33}}{4} when ± is minus. Subtract 10\sqrt{33} from -30.

$$x=\frac{-10\sqrt{33}-30}{4}$$

Divide -30-10\sqrt{33} by 4.

$$x=\frac{-5\sqrt{33}-15}{2}$$

The equation is now solved.

$$x=\frac{5\sqrt{33}-15}{2} x=\frac{-5\sqrt{33}-15}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$30x+2x^{2}=300$$

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.

$$2x^{2}+30x=300$$

Divide both sides by 2.

$$\frac{2x^{2}+30x}{2}=\frac{300}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{30}{2}x=\frac{300}{2}$$

Divide 30 by 2.

$$x^{2}+15x=\frac{300}{2}$$

Divide 300 by 2.

$$x^{2}+15x=150$$

Divide 15, the coefficient of the x term, by 2 to get \frac{15}{2}. Then add the square of \frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=150+\left(\frac{15}{2}\right)^{2}$$

Square \frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+15x+\frac{225}{4}=150+\frac{225}{4}$$

Add 150 to \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{825}{4}$$

Factor x^{2}+15x+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{825}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{825}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{5\sqrt{33}}{2} x+\frac{15}{2}=-\frac{5\sqrt{33}}{2}$$

Subtract \frac{15}{2} from both sides of the equation.

$$x=\frac{5\sqrt{33}-15}{2} x=\frac{-5\sqrt{33}-15}{2}$$