Solve 5(x+3)^2-3x^2=335

$$5 ( x + 3 ) ^ { 2 } - 3 x ^ { 2 } = 335$$

Solve for x

$$x = \frac{\sqrt{805} - 15}{2} \approx 6.686260959$$

$$x=\frac{-\sqrt{805}-15}{2}\approx -21.686260959$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$5\left(x^{2}+6x+9\right)-3x^{2}=335$$

Use the distributive property to multiply 5 by x^{2}+6x+9.

$$5x^{2}+30x+45-3x^{2}=335$$

Combine 5x^{2} and -3x^{2} to get 2x^{2}.

$$2x^{2}+30x+45=335$$

Subtract 335 from both sides.

$$2x^{2}+30x+45-335=0$$

Subtract 335 from 45 to get -290.

$$2x^{2}+30x-290=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 30 for b, and -290 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 2\left(-290\right)}}{2\times 2}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 2\left(-290\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-30±\sqrt{900-8\left(-290\right)}}{2\times 2}$$

Multiply -8 times -290.

$$x=\frac{-30±\sqrt{900+2320}}{2\times 2}$$

Add 900 to 2320.

$$x=\frac{-30±\sqrt{3220}}{2\times 2}$$

Take the square root of 3220.

$$x=\frac{-30±2\sqrt{805}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-30±2\sqrt{805}}{4}$$

Now solve the equation x=\frac{-30±2\sqrt{805}}{4} when ± is plus. Add -30 to 2\sqrt{805}.

$$x=\frac{2\sqrt{805}-30}{4}$$

Divide -30+2\sqrt{805} by 4.

$$x=\frac{\sqrt{805}-15}{2}$$

Now solve the equation x=\frac{-30±2\sqrt{805}}{4} when ± is minus. Subtract 2\sqrt{805} from -30.

$$x=\frac{-2\sqrt{805}-30}{4}$$

Divide -30-2\sqrt{805} by 4.

$$x=\frac{-\sqrt{805}-15}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{805}-15}{2} x=\frac{-\sqrt{805}-15}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+3\right)^{2}.

$$5\left(x^{2}+6x+9\right)-3x^{2}=335$$

Use the distributive property to multiply 5 by x^{2}+6x+9.

$$5x^{2}+30x+45-3x^{2}=335$$

Combine 5x^{2} and -3x^{2} to get 2x^{2}.

$$2x^{2}+30x+45=335$$

Subtract 45 from both sides.

$$2x^{2}+30x=335-45$$

Subtract 45 from 335 to get 290.

$$2x^{2}+30x=290$$

Divide both sides by 2.

$$\frac{2x^{2}+30x}{2}=\frac{290}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{30}{2}x=\frac{290}{2}$$

Divide 30 by 2.

$$x^{2}+15x=\frac{290}{2}$$

Divide 290 by 2.

$$x^{2}+15x=145$$

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.

$$x^{2}+15x+\left(\frac{15}{2}\right)^{2}=145+\left(\frac{15}{2}\right)^{2}$$

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

$$x^{2}+15x+\frac{225}{4}=145+\frac{225}{4}$$

Add 145 to \frac{225}{4}.

$$x^{2}+15x+\frac{225}{4}=\frac{805}{4}$$

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}.

$$\left(x+\frac{15}{2}\right)^{2}=\frac{805}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{15}{2}\right)^{2}}=\sqrt{\frac{805}{4}}$$

Simplify.

$$x+\frac{15}{2}=\frac{\sqrt{805}}{2} x+\frac{15}{2}=-\frac{\sqrt{805}}{2}$$

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

$$x=\frac{\sqrt{805}-15}{2} x=\frac{-\sqrt{805}-15}{2}$$