Solve 15-x^2+4x-5=0

$$15 - x ^ { 2 } + 4 x - 5 = 0$$

Solve for x

$$x=\sqrt{14}+2\approx 5.741657387$$

$$x=2-\sqrt{14}\approx -1.741657387$$

Steps Using the Quadratic Formula

Subtract 5 from 15 to get 10.

$$10-x^{2}+4x=0$$

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.

$$-x^{2}+4x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 4 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\left(-1\right)\times 10}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-4±\sqrt{16+4\times 10}}{2\left(-1\right)}$$

Multiply 4 times 10.

$$x=\frac{-4±\sqrt{16+40}}{2\left(-1\right)}$$

Add 16 to 40.

$$x=\frac{-4±\sqrt{56}}{2\left(-1\right)}$$

Take the square root of 56.

$$x=\frac{-4±2\sqrt{14}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{-4±2\sqrt{14}}{-2}$$

Now solve the equation x=\frac{-4±2\sqrt{14}}{-2} when ± is plus. Add -4 to 2\sqrt{14}.

$$x=\frac{2\sqrt{14}-4}{-2}$$

Divide -4+2\sqrt{14} by -2.

$$x=2-\sqrt{14}$$

Now solve the equation x=\frac{-4±2\sqrt{14}}{-2} when ± is minus. Subtract 2\sqrt{14} from -4.

$$x=\frac{-2\sqrt{14}-4}{-2}$$

Divide -4-2\sqrt{14} by -2.

$$x=\sqrt{14}+2$$

The equation is now solved.

$$x=2-\sqrt{14} x=\sqrt{14}+2$$

Steps for Completing the Square

Subtract 5 from 15 to get 10.

$$10-x^{2}+4x=0$$

Subtract 10 from both sides. Anything subtracted from zero gives its negation.

$$-x^{2}+4x=-10$$

Divide both sides by -1.

$$\frac{-x^{2}+4x}{-1}=-\frac{10}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\frac{4}{-1}x=-\frac{10}{-1}$$

Divide 4 by -1.

$$x^{2}-4x=-\frac{10}{-1}$$

Divide -10 by -1.

$$x^{2}-4x=10$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=10+\left(-2\right)^{2}$$

Square -2.

$$x^{2}-4x+4=10+4$$

Add 10 to 4.

$$x^{2}-4x+4=14$$

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}.

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

Take the square root of both sides of the equation.

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

Simplify.

$$x-2=\sqrt{14} x-2=-\sqrt{14}$$

Add 2 to both sides of the equation.

$$x=\sqrt{14}+2 x=2-\sqrt{14}$$