Solve 5*(x+3)x=7

$$5 \times (x+3)x=7$$

Solve for x

$$x=\frac{\sqrt{365}}{10}-\frac{3}{2}\approx 0.410497317$$

$$x=-\frac{\sqrt{365}}{10}-\frac{3}{2}\approx -3.410497317$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 5 by x+3.

$$\left(5x+15\right)x=7$$

Use the distributive property to multiply 5x+15 by x.

$$5x^{2}+15x=7$$

Subtract 7 from both sides.

$$5x^{2}+15x-7=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 15 for b, and -7 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-15±\sqrt{15^{2}-4\times 5\left(-7\right)}}{2\times 5}$$

Square 15.

$$x=\frac{-15±\sqrt{225-4\times 5\left(-7\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-15±\sqrt{225-20\left(-7\right)}}{2\times 5}$$

Multiply -20 times -7.

$$x=\frac{-15±\sqrt{225+140}}{2\times 5}$$

Add 225 to 140.

$$x=\frac{-15±\sqrt{365}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-15±\sqrt{365}}{10}$$

Now solve the equation x=\frac{-15±\sqrt{365}}{10} when ± is plus. Add -15 to \sqrt{365}.

$$x=\frac{\sqrt{365}-15}{10}$$

Divide -15+\sqrt{365} by 10.

$$x=\frac{\sqrt{365}}{10}-\frac{3}{2}$$

Now solve the equation x=\frac{-15±\sqrt{365}}{10} when ± is minus. Subtract \sqrt{365} from -15.

$$x=\frac{-\sqrt{365}-15}{10}$$

Divide -15-\sqrt{365} by 10.

$$x=-\frac{\sqrt{365}}{10}-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{365}}{10}-\frac{3}{2} x=-\frac{\sqrt{365}}{10}-\frac{3}{2}$$

Steps for Completing the Square

Use the distributive property to multiply 5 by x+3.

$$\left(5x+15\right)x=7$$

Use the distributive property to multiply 5x+15 by x.

$$5x^{2}+15x=7$$

Divide both sides by 5.

$$\frac{5x^{2}+15x}{5}=\frac{7}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{15}{5}x=\frac{7}{5}$$

Divide 15 by 5.

$$x^{2}+3x=\frac{7}{5}$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{7}{5}+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=\frac{7}{5}+\frac{9}{4}$$

Add \frac{7}{5} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=\frac{73}{20}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{73}{20}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{73}{20}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{365}}{10} x+\frac{3}{2}=-\frac{\sqrt{365}}{10}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{365}}{10}-\frac{3}{2} x=-\frac{\sqrt{365}}{10}-\frac{3}{2}$$