Solve s^2-26s=-47

$$s ^ { 2 } - 26 s = - 47$$

Solve for s

$$s=\sqrt{122}+13\approx 24.045361017$$

$$s=13-\sqrt{122}\approx 1.954638983$$

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.

$$s^{2}-26s=-47$$

Add 47 to both sides of the equation.

$$s^{2}-26s-\left(-47\right)=-47-\left(-47\right)$$

Subtracting -47 from itself leaves 0.

$$s^{2}-26s-\left(-47\right)=0$$

Subtract -47 from 0.

$$s^{2}-26s+47=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -26 for b, and 47 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 47}}{2}$$

Square -26.

$$s=\frac{-\left(-26\right)±\sqrt{676-4\times 47}}{2}$$

Multiply -4 times 47.

$$s=\frac{-\left(-26\right)±\sqrt{676-188}}{2}$$

Add 676 to -188.

$$s=\frac{-\left(-26\right)±\sqrt{488}}{2}$$

Take the square root of 488.

$$s=\frac{-\left(-26\right)±2\sqrt{122}}{2}$$

The opposite of -26 is 26.

$$s=\frac{26±2\sqrt{122}}{2}$$

Now solve the equation s=\frac{26±2\sqrt{122}}{2} when ± is plus. Add 26 to 2\sqrt{122}.

$$s=\frac{2\sqrt{122}+26}{2}$$

Divide 26+2\sqrt{122} by 2.

$$s=\sqrt{122}+13$$

Now solve the equation s=\frac{26±2\sqrt{122}}{2} when ± is minus. Subtract 2\sqrt{122} from 26.

$$s=\frac{26-2\sqrt{122}}{2}$$

Divide 26-2\sqrt{122} by 2.

$$s=13-\sqrt{122}$$

The equation is now solved.

$$s=\sqrt{122}+13 s=13-\sqrt{122}$$

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.

$$s^{2}-26s=-47$$

Divide -26, the coefficient of the x term, by 2 to get -13. Then add the square of -13 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$s^{2}-26s+\left(-13\right)^{2}=-47+\left(-13\right)^{2}$$

Square -13.

$$s^{2}-26s+169=-47+169$$

Add -47 to 169.

$$s^{2}-26s+169=122$$

Factor s^{2}-26s+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}.

$$\left(s-13\right)^{2}=122$$

Take the square root of both sides of the equation.

$$\sqrt{\left(s-13\right)^{2}}=\sqrt{122}$$

Simplify.

$$s-13=\sqrt{122} s-13=-\sqrt{122}$$

Add 13 to both sides of the equation.

$$s=\sqrt{122}+13 s=13-\sqrt{122}$$