Solve -15=20x-5x^2

$$- 15 = 20 x - 5 x ^ { 2 }$$

Solve for x

$$x=\sqrt{7}+2\approx 4.645751311$$

$$x=2-\sqrt{7}\approx -0.645751311$$

Steps Using the Quadratic Formula

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

$$20x-5x^{2}=-15$$

Add 15 to both sides.

$$20x-5x^{2}+15=0$$

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.

$$-5x^{2}+20x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 20 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-5\right)\times 15}}{2\left(-5\right)}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-5\right)\times 15}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$x=\frac{-20±\sqrt{400+20\times 15}}{2\left(-5\right)}$$

Multiply 20 times 15.

$$x=\frac{-20±\sqrt{400+300}}{2\left(-5\right)}$$

Add 400 to 300.

$$x=\frac{-20±\sqrt{700}}{2\left(-5\right)}$$

Take the square root of 700.

$$x=\frac{-20±10\sqrt{7}}{2\left(-5\right)}$$

Multiply 2 times -5.

$$x=\frac{-20±10\sqrt{7}}{-10}$$

Now solve the equation x=\frac{-20±10\sqrt{7}}{-10} when ± is plus. Add -20 to 10\sqrt{7}.

$$x=\frac{10\sqrt{7}-20}{-10}$$

Divide -20+10\sqrt{7} by -10.

$$x=2-\sqrt{7}$$

Now solve the equation x=\frac{-20±10\sqrt{7}}{-10} when ± is minus. Subtract 10\sqrt{7} from -20.

$$x=\frac{-10\sqrt{7}-20}{-10}$$

Divide -20-10\sqrt{7} by -10.

$$x=\sqrt{7}+2$$

The equation is now solved.

$$x=2-\sqrt{7} x=\sqrt{7}+2$$

Steps for Completing the Square

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

$$20x-5x^{2}=-15$$

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.

$$-5x^{2}+20x=-15$$

Divide both sides by -5.

$$\frac{-5x^{2}+20x}{-5}=-\frac{15}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$x^{2}+\frac{20}{-5}x=-\frac{15}{-5}$$

Divide 20 by -5.

$$x^{2}-4x=-\frac{15}{-5}$$

Divide -15 by -5.

$$x^{2}-4x=3$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-4x+\left(-2\right)^{2}=3+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=3+4$$

Add 3 to 4.

$$x^{2}-4x+4=7$$

Factor x^{2}-4x+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}.

$$\left(x-2\right)^{2}=7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{7}$$

Simplify.

$$x-2=\sqrt{7} x-2=-\sqrt{7}$$

Add 2 to both sides of the equation.

$$x=\sqrt{7}+2 x=2-\sqrt{7}$$