Solve 25=30x+5x^2

$$25 = 30x+5 { x }^{ 2 }$$

Solve for x (complex solution)

$$x=\sqrt{14}-3\approx 0.741657387$$

$$x=-\left(\sqrt{14}+3\right)\approx -6.741657387$$

Steps Using the Quadratic Formula

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

$$30x+5x^{2}=25$$

Subtract 25 from both sides.

$$30x+5x^{2}-25=0$$

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.

$$5x^{2}+30x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 30 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 5\left(-25\right)}}{2\times 5}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 5\left(-25\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-30±\sqrt{900-20\left(-25\right)}}{2\times 5}$$

Multiply -20 times -25.

$$x=\frac{-30±\sqrt{900+500}}{2\times 5}$$

Add 900 to 500.

$$x=\frac{-30±\sqrt{1400}}{2\times 5}$$

Take the square root of 1400.

$$x=\frac{-30±10\sqrt{14}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-30±10\sqrt{14}}{10}$$

Now solve the equation x=\frac{-30±10\sqrt{14}}{10} when ± is plus. Add -30 to 10\sqrt{14}.

$$x=\frac{10\sqrt{14}-30}{10}$$

Divide -30+10\sqrt{14} by 10.

$$x=\sqrt{14}-3$$

Now solve the equation x=\frac{-30±10\sqrt{14}}{10} when ± is minus. Subtract 10\sqrt{14} from -30.

$$x=\frac{-10\sqrt{14}-30}{10}$$

Divide -30-10\sqrt{14} by 10.

$$x=-\sqrt{14}-3$$

The equation is now solved.

$$x=\sqrt{14}-3 x=-\sqrt{14}-3$$

Steps for Completing the Square

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

$$30x+5x^{2}=25$$

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.

$$5x^{2}+30x=25$$

Divide both sides by 5.

$$\frac{5x^{2}+30x}{5}=\frac{25}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{30}{5}x=\frac{25}{5}$$

Divide 30 by 5.

$$x^{2}+6x=\frac{25}{5}$$

Divide 25 by 5.

$$x^{2}+6x=5$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=5+3^{2}$$

Square 3.

$$x^{2}+6x+9=5+9$$

Add 5 to 9.

$$x^{2}+6x+9=14$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{14}$$

Simplify.

$$x+3=\sqrt{14} x+3=-\sqrt{14}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{14}-3 x=-\sqrt{14}-3$$

Solve for x

$$x=\sqrt{14}-3\approx 0.741657387$$

$$x=-\sqrt{14}-3\approx -6.741657387$$

Steps Using the Quadratic Formula

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

$$30x+5x^{2}=25$$

Subtract 25 from both sides.

$$30x+5x^{2}-25=0$$

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.

$$5x^{2}+30x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 30 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-30±\sqrt{30^{2}-4\times 5\left(-25\right)}}{2\times 5}$$

Square 30.

$$x=\frac{-30±\sqrt{900-4\times 5\left(-25\right)}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-30±\sqrt{900-20\left(-25\right)}}{2\times 5}$$

Multiply -20 times -25.

$$x=\frac{-30±\sqrt{900+500}}{2\times 5}$$

Add 900 to 500.

$$x=\frac{-30±\sqrt{1400}}{2\times 5}$$

Take the square root of 1400.

$$x=\frac{-30±10\sqrt{14}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-30±10\sqrt{14}}{10}$$

Now solve the equation x=\frac{-30±10\sqrt{14}}{10} when ± is plus. Add -30 to 10\sqrt{14}.

$$x=\frac{10\sqrt{14}-30}{10}$$

Divide -30+10\sqrt{14} by 10.

$$x=\sqrt{14}-3$$

Now solve the equation x=\frac{-30±10\sqrt{14}}{10} when ± is minus. Subtract 10\sqrt{14} from -30.

$$x=\frac{-10\sqrt{14}-30}{10}$$

Divide -30-10\sqrt{14} by 10.

$$x=-\sqrt{14}-3$$

The equation is now solved.

$$x=\sqrt{14}-3 x=-\sqrt{14}-3$$

Steps for Completing the Square

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

$$30x+5x^{2}=25$$

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.

$$5x^{2}+30x=25$$

Divide both sides by 5.

$$\frac{5x^{2}+30x}{5}=\frac{25}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{30}{5}x=\frac{25}{5}$$

Divide 30 by 5.

$$x^{2}+6x=\frac{25}{5}$$

Divide 25 by 5.

$$x^{2}+6x=5$$

Divide 6, the coefficient of the x term, by 2 to get 3. Then add the square of 3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+6x+3^{2}=5+3^{2}$$

Square 3.

$$x^{2}+6x+9=5+9$$

Add 5 to 9.

$$x^{2}+6x+9=14$$

Factor x^{2}+6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+3\right)^{2}=14$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{14}$$

Simplify.

$$x+3=\sqrt{14} x+3=-\sqrt{14}$$

Subtract 3 from both sides of the equation.

$$x=\sqrt{14}-3 x=-\sqrt{14}-3$$