Solve 1=11+17x^2-32x

$$1 = 11 + 17 x ^ { 2 } - 32 x$$

Solve for x

$$x = \frac{\sqrt{86} + 16}{17} \approx 1.486683441$$

$$x=\frac{16-\sqrt{86}}{17}\approx 0.3956695$$

Steps Using the Quadratic Formula

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

$$11+17x^{2}-32x=1$$

Subtract 1 from both sides.

$$11+17x^{2}-32x-1=0$$

Subtract 1 from 11 to get 10.

$$10+17x^{2}-32x=0$$

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.

$$17x^{2}-32x+10=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 17 for a, -32 for b, and 10 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 17\times 10}}{2\times 17}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 17\times 10}}{2\times 17}$$

Multiply -4 times 17.

$$x=\frac{-\left(-32\right)±\sqrt{1024-68\times 10}}{2\times 17}$$

Multiply -68 times 10.

$$x=\frac{-\left(-32\right)±\sqrt{1024-680}}{2\times 17}$$

Add 1024 to -680.

$$x=\frac{-\left(-32\right)±\sqrt{344}}{2\times 17}$$

Take the square root of 344.

$$x=\frac{-\left(-32\right)±2\sqrt{86}}{2\times 17}$$

The opposite of -32 is 32.

$$x=\frac{32±2\sqrt{86}}{2\times 17}$$

Multiply 2 times 17.

$$x=\frac{32±2\sqrt{86}}{34}$$

Now solve the equation x=\frac{32±2\sqrt{86}}{34} when ± is plus. Add 32 to 2\sqrt{86}.

$$x=\frac{2\sqrt{86}+32}{34}$$

Divide 32+2\sqrt{86} by 34.

$$x=\frac{\sqrt{86}+16}{17}$$

Now solve the equation x=\frac{32±2\sqrt{86}}{34} when ± is minus. Subtract 2\sqrt{86} from 32.

$$x=\frac{32-2\sqrt{86}}{34}$$

Divide 32-2\sqrt{86} by 34.

$$x=\frac{16-\sqrt{86}}{17}$$

The equation is now solved.

$$x=\frac{\sqrt{86}+16}{17} x=\frac{16-\sqrt{86}}{17}$$

Steps for Completing the Square

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

$$11+17x^{2}-32x=1$$

Subtract 11 from both sides.

$$17x^{2}-32x=1-11$$

Subtract 11 from 1 to get -10.

$$17x^{2}-32x=-10$$

Divide both sides by 17.

$$\frac{17x^{2}-32x}{17}=-\frac{10}{17}$$

Dividing by 17 undoes the multiplication by 17.

$$x^{2}-\frac{32}{17}x=-\frac{10}{17}$$

Divide -\frac{32}{17}, the coefficient of the x term, by 2 to get -\frac{16}{17}. Then add the square of -\frac{16}{17} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{32}{17}x+\left(-\frac{16}{17}\right)^{2}=-\frac{10}{17}+\left(-\frac{16}{17}\right)^{2}$$

Square -\frac{16}{17} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{32}{17}x+\frac{256}{289}=-\frac{10}{17}+\frac{256}{289}$$

Add -\frac{10}{17} to \frac{256}{289} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{32}{17}x+\frac{256}{289}=\frac{86}{289}$$

Factor x^{2}-\frac{32}{17}x+\frac{256}{289}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{16}{17}\right)^{2}=\frac{86}{289}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{16}{17}\right)^{2}}=\sqrt{\frac{86}{289}}$$

Simplify.

$$x-\frac{16}{17}=\frac{\sqrt{86}}{17} x-\frac{16}{17}=-\frac{\sqrt{86}}{17}$$

Add \frac{16}{17} to both sides of the equation.

$$x=\frac{\sqrt{86}+16}{17} x=\frac{16-\sqrt{86}}{17}$$