Solve 2x^2+3x=-3x^2+9

$$2 x ^ { 2 } + 3 x = - 3 x ^ { 2 } + 9$$

Solve for x

$$x = \frac{3 \sqrt{21} - 3}{10} \approx 1.074772708$$

$$x=\frac{-3\sqrt{21}-3}{10}\approx -1.674772708$$

Steps Using the Quadratic Formula

Add 3x^{2} to both sides.

$$2x^{2}+3x+3x^{2}=9$$

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

$$5x^{2}+3x=9$$

Subtract 9 from both sides.

$$5x^{2}+3x-9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 3 for b, and -9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\times 5\left(-9\right)}}{2\times 5}$$

Square 3.

$$x=\frac{-3±\sqrt{9-4\times 5\left(-9\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-3±\sqrt{9-20\left(-9\right)}}{2\times 5}$$

Multiply -20 times -9.

$$x=\frac{-3±\sqrt{9+180}}{2\times 5}$$

Add 9 to 180.

$$x=\frac{-3±\sqrt{189}}{2\times 5}$$

Take the square root of 189.

$$x=\frac{-3±3\sqrt{21}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-3±3\sqrt{21}}{10}$$

Now solve the equation x=\frac{-3±3\sqrt{21}}{10} when ± is plus. Add -3 to 3\sqrt{21}.

$$x=\frac{3\sqrt{21}-3}{10}$$

Now solve the equation x=\frac{-3±3\sqrt{21}}{10} when ± is minus. Subtract 3\sqrt{21} from -3.

$$x=\frac{-3\sqrt{21}-3}{10}$$

The equation is now solved.

$$x=\frac{3\sqrt{21}-3}{10} x=\frac{-3\sqrt{21}-3}{10}$$

Steps for Completing the Square

Add 3x^{2} to both sides.

$$2x^{2}+3x+3x^{2}=9$$

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

$$5x^{2}+3x=9$$

Divide both sides by 5.

$$\frac{5x^{2}+3x}{5}=\frac{9}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{3}{5}x=\frac{9}{5}$$

Divide \frac{3}{5}, the coefficient of the x term, by 2 to get \frac{3}{10}. Then add the square of \frac{3}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{5}x+\left(\frac{3}{10}\right)^{2}=\frac{9}{5}+\left(\frac{3}{10}\right)^{2}$$

Square \frac{3}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=\frac{9}{5}+\frac{9}{100}$$

Add \frac{9}{5} to \frac{9}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=\frac{189}{100}$$

Factor x^{2}+\frac{3}{5}x+\frac{9}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{10}\right)^{2}=\frac{189}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{10}\right)^{2}}=\sqrt{\frac{189}{100}}$$

Simplify.

$$x+\frac{3}{10}=\frac{3\sqrt{21}}{10} x+\frac{3}{10}=-\frac{3\sqrt{21}}{10}$$

Subtract \frac{3}{10} from both sides of the equation.

$$x=\frac{3\sqrt{21}-3}{10} x=\frac{-3\sqrt{21}-3}{10}$$