Solve 13x^2-32x+1=0

$$13 { x }^{ 2 } -32x+1=0$$

Solve for x

$$x = \frac{9 \sqrt{3} + 16}{13} \approx 2.429881328$$

$$x=\frac{16-9\sqrt{3}}{13}\approx 0.031657133$$

Steps Using the Quadratic Formula

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.

$$13x^{2}-32x+1=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 13 for a, -32 for b, and 1 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 13}}{2\times 13}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\times 13}}{2\times 13}$$

Multiply -4 times 13.

$$x=\frac{-\left(-32\right)±\sqrt{1024-52}}{2\times 13}$$

Add 1024 to -52.

$$x=\frac{-\left(-32\right)±\sqrt{972}}{2\times 13}$$

Take the square root of 972.

$$x=\frac{-\left(-32\right)±18\sqrt{3}}{2\times 13}$$

The opposite of -32 is 32.

$$x=\frac{32±18\sqrt{3}}{2\times 13}$$

Multiply 2 times 13.

$$x=\frac{32±18\sqrt{3}}{26}$$

Now solve the equation x=\frac{32±18\sqrt{3}}{26} when ± is plus. Add 32 to 18\sqrt{3}.

$$x=\frac{18\sqrt{3}+32}{26}$$

Divide 32+18\sqrt{3} by 26.

$$x=\frac{9\sqrt{3}+16}{13}$$

Now solve the equation x=\frac{32±18\sqrt{3}}{26} when ± is minus. Subtract 18\sqrt{3} from 32.

$$x=\frac{32-18\sqrt{3}}{26}$$

Divide 32-18\sqrt{3} by 26.

$$x=\frac{16-9\sqrt{3}}{13}$$

The equation is now solved.

$$x=\frac{9\sqrt{3}+16}{13} x=\frac{16-9\sqrt{3}}{13}$$

Steps for Completing the Square

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.

$$13x^{2}-32x+1=0$$

Subtract 1 from both sides of the equation.

$$13x^{2}-32x+1-1=-1$$

Subtracting 1 from itself leaves 0.

$$13x^{2}-32x=-1$$

Divide both sides by 13.

$$\frac{13x^{2}-32x}{13}=-\frac{1}{13}$$

Dividing by 13 undoes the multiplication by 13.

$$x^{2}-\frac{32}{13}x=-\frac{1}{13}$$

Divide -\frac{32}{13}, the coefficient of the x term, by 2 to get -\frac{16}{13}. Then add the square of -\frac{16}{13} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{32}{13}x+\left(-\frac{16}{13}\right)^{2}=-\frac{1}{13}+\left(-\frac{16}{13}\right)^{2}$$

Square -\frac{16}{13} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{32}{13}x+\frac{256}{169}=-\frac{1}{13}+\frac{256}{169}$$

Add -\frac{1}{13} to \frac{256}{169} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{32}{13}x+\frac{256}{169}=\frac{243}{169}$$

Factor x^{2}-\frac{32}{13}x+\frac{256}{169}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{16}{13}\right)^{2}=\frac{243}{169}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{16}{13}\right)^{2}}=\sqrt{\frac{243}{169}}$$

Simplify.

$$x-\frac{16}{13}=\frac{9\sqrt{3}}{13} x-\frac{16}{13}=-\frac{9\sqrt{3}}{13}$$

Add \frac{16}{13} to both sides of the equation.

$$x=\frac{9\sqrt{3}+16}{13} x=\frac{16-9\sqrt{3}}{13}$$