Solve x=x^2+17x+3

$$x = x ^ { 2 } + 17 x + 3$$

Solve for x (complex solution)

$$x=\sqrt{61}-8\approx -0.189750324$$

$$x=-\left(\sqrt{61}+8\right)\approx -15.810249676$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$x-x^{2}=17x+3$$

Subtract 17x from both sides.

$$x-x^{2}-17x=3$$

Combine x and -17x to get -16x.

$$-16x-x^{2}=3$$

Subtract 3 from both sides.

$$-16x-x^{2}-3=0$$

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.

$$-x^{2}-16x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -16 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-16\right)±\sqrt{256+4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$x=\frac{-\left(-16\right)±\sqrt{256-12}}{2\left(-1\right)}$$

Add 256 to -12.

$$x=\frac{-\left(-16\right)±\sqrt{244}}{2\left(-1\right)}$$

Take the square root of 244.

$$x=\frac{-\left(-16\right)±2\sqrt{61}}{2\left(-1\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{61}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{16±2\sqrt{61}}{-2}$$

Now solve the equation x=\frac{16±2\sqrt{61}}{-2} when ± is plus. Add 16 to 2\sqrt{61}.

$$x=\frac{2\sqrt{61}+16}{-2}$$

Divide 16+2\sqrt{61} by -2.

$$x=-\left(\sqrt{61}+8\right)$$

Now solve the equation x=\frac{16±2\sqrt{61}}{-2} when ± is minus. Subtract 2\sqrt{61} from 16.

$$x=\frac{16-2\sqrt{61}}{-2}$$

Divide 16-2\sqrt{61} by -2.

$$x=\sqrt{61}-8$$

The equation is now solved.

$$x=-\left(\sqrt{61}+8\right) x=\sqrt{61}-8$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$x-x^{2}=17x+3$$

Subtract 17x from both sides.

$$x-x^{2}-17x=3$$

Combine x and -17x to get -16x.

$$-16x-x^{2}=3$$

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.

$$-x^{2}-16x=3$$

Divide both sides by -1.

$$\frac{-x^{2}-16x}{-1}=\frac{3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{16}{-1}\right)x=\frac{3}{-1}$$

Divide -16 by -1.

$$x^{2}+16x=\frac{3}{-1}$$

Divide 3 by -1.

$$x^{2}+16x=-3$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+16x+8^{2}=-3+8^{2}$$

Square 8.

$$x^{2}+16x+64=-3+64$$

Add -3 to 64.

$$x^{2}+16x+64=61$$

Factor x^{2}+16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+8\right)^{2}=61$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{61}$$

Simplify.

$$x+8=\sqrt{61} x+8=-\sqrt{61}$$

Subtract 8 from both sides of the equation.

$$x=\sqrt{61}-8 x=-\sqrt{61}-8$$

Solve for x

$$x=\sqrt{61}-8\approx -0.189750324$$

$$x=-\sqrt{61}-8\approx -15.810249676$$

Steps Using the Quadratic Formula

Subtract x^{2} from both sides.

$$x-x^{2}=17x+3$$

Subtract 17x from both sides.

$$x-x^{2}-17x=3$$

Combine x and -17x to get -16x.

$$-16x-x^{2}=3$$

Subtract 3 from both sides.

$$-16x-x^{2}-3=0$$

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.

$$-x^{2}-16x-3=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -16 for b, and -3 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\left(-1\right)\left(-3\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{-\left(-16\right)±\sqrt{256+4\left(-3\right)}}{2\left(-1\right)}$$

Multiply 4 times -3.

$$x=\frac{-\left(-16\right)±\sqrt{256-12}}{2\left(-1\right)}$$

Add 256 to -12.

$$x=\frac{-\left(-16\right)±\sqrt{244}}{2\left(-1\right)}$$

Take the square root of 244.

$$x=\frac{-\left(-16\right)±2\sqrt{61}}{2\left(-1\right)}$$

The opposite of -16 is 16.

$$x=\frac{16±2\sqrt{61}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{16±2\sqrt{61}}{-2}$$

Now solve the equation x=\frac{16±2\sqrt{61}}{-2} when ± is plus. Add 16 to 2\sqrt{61}.

$$x=\frac{2\sqrt{61}+16}{-2}$$

Divide 16+2\sqrt{61} by -2.

$$x=-\left(\sqrt{61}+8\right)$$

Now solve the equation x=\frac{16±2\sqrt{61}}{-2} when ± is minus. Subtract 2\sqrt{61} from 16.

$$x=\frac{16-2\sqrt{61}}{-2}$$

Divide 16-2\sqrt{61} by -2.

$$x=\sqrt{61}-8$$

The equation is now solved.

$$x=-\left(\sqrt{61}+8\right) x=\sqrt{61}-8$$

Steps for Completing the Square

Subtract x^{2} from both sides.

$$x-x^{2}=17x+3$$

Subtract 17x from both sides.

$$x-x^{2}-17x=3$$

Combine x and -17x to get -16x.

$$-16x-x^{2}=3$$

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.

$$-x^{2}-16x=3$$

Divide both sides by -1.

$$\frac{-x^{2}-16x}{-1}=\frac{3}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$x^{2}+\left(-\frac{16}{-1}\right)x=\frac{3}{-1}$$

Divide -16 by -1.

$$x^{2}+16x=\frac{3}{-1}$$

Divide 3 by -1.

$$x^{2}+16x=-3$$

Divide 16, the coefficient of the x term, by 2 to get 8. Then add the square of 8 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+16x+8^{2}=-3+8^{2}$$

Square 8.

$$x^{2}+16x+64=-3+64$$

Add -3 to 64.

$$x^{2}+16x+64=61$$

Factor x^{2}+16x+64. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+8\right)^{2}=61$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+8\right)^{2}}=\sqrt{61}$$

Simplify.

$$x+8=\sqrt{61} x+8=-\sqrt{61}$$

Subtract 8 from both sides of the equation.

$$x=\sqrt{61}-8 x=-\sqrt{61}-8$$