Solve 26x^2-156x+205=0

$$26 { x }^{ 2 } -156x+205=0$$

Solve for x

$$x=\frac{\sqrt{754}}{26}+3\approx 4.056117709$$

$$x=-\frac{\sqrt{754}}{26}+3\approx 1.943882291$$

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.

$$26x^{2}-156x+205=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 26 for a, -156 for b, and 205 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-156\right)±\sqrt{\left(-156\right)^{2}-4\times 26\times 205}}{2\times 26}$$

Square -156.

$$x=\frac{-\left(-156\right)±\sqrt{24336-4\times 26\times 205}}{2\times 26}$$

Multiply -4 times 26.

$$x=\frac{-\left(-156\right)±\sqrt{24336-104\times 205}}{2\times 26}$$

Multiply -104 times 205.

$$x=\frac{-\left(-156\right)±\sqrt{24336-21320}}{2\times 26}$$

Add 24336 to -21320.

$$x=\frac{-\left(-156\right)±\sqrt{3016}}{2\times 26}$$

Take the square root of 3016.

$$x=\frac{-\left(-156\right)±2\sqrt{754}}{2\times 26}$$

The opposite of -156 is 156.

$$x=\frac{156±2\sqrt{754}}{2\times 26}$$

Multiply 2 times 26.

$$x=\frac{156±2\sqrt{754}}{52}$$

Now solve the equation x=\frac{156±2\sqrt{754}}{52} when ± is plus. Add 156 to 2\sqrt{754}.

$$x=\frac{2\sqrt{754}+156}{52}$$

Divide 156+2\sqrt{754} by 52.

$$x=\frac{\sqrt{754}}{26}+3$$

Now solve the equation x=\frac{156±2\sqrt{754}}{52} when ± is minus. Subtract 2\sqrt{754} from 156.

$$x=\frac{156-2\sqrt{754}}{52}$$

Divide 156-2\sqrt{754} by 52.

$$x=-\frac{\sqrt{754}}{26}+3$$

The equation is now solved.

$$x=\frac{\sqrt{754}}{26}+3 x=-\frac{\sqrt{754}}{26}+3$$

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.

$$26x^{2}-156x+205=0$$

Subtract 205 from both sides of the equation.

$$26x^{2}-156x+205-205=-205$$

Subtracting 205 from itself leaves 0.

$$26x^{2}-156x=-205$$

Divide both sides by 26.

$$\frac{26x^{2}-156x}{26}=-\frac{205}{26}$$

Dividing by 26 undoes the multiplication by 26.

$$x^{2}+\left(-\frac{156}{26}\right)x=-\frac{205}{26}$$

Divide -156 by 26.

$$x^{2}-6x=-\frac{205}{26}$$

Divide -6, the coefficient of the x term, by 2 to get -3. Then add the square of -3 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-6x+\left(-3\right)^{2}=-\frac{205}{26}+\left(-3\right)^{2}$$

Square -3.

$$x^{2}-6x+9=-\frac{205}{26}+9$$

Add -\frac{205}{26} to 9.

$$x^{2}-6x+9=\frac{29}{26}$$

Factor x^{2}-6x+9. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-3\right)^{2}=\frac{29}{26}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{\frac{29}{26}}$$

Simplify.

$$x-3=\frac{\sqrt{754}}{26} x-3=-\frac{\sqrt{754}}{26}$$

Add 3 to both sides of the equation.

$$x=\frac{\sqrt{754}}{26}+3 x=-\frac{\sqrt{754}}{26}+3$$