Solve s^2-24s+169=25

$$s ^ { 2 } - 24 s + 169 = 25$$

Solve for s

$$s=12$$

Steps Using Factoring

Subtract 25 from both sides.

$$s^{2}-24s+169-25=0$$

Subtract 25 from 169 to get 144.

$$s^{2}-24s+144=0$$

To solve the equation, factor s^{2}-24s+144 using formula s^{2}+\left(a+b\right)s+ab=\left(s+a\right)\left(s+b\right). To find a and b, set up a system to be solved.

$$a+b=-24 ab=144$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Calculate the sum for each pair.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

The solution is the pair that gives sum -24.

$$a=-12 b=-12$$

Rewrite factored expression \left(s+a\right)\left(s+b\right) using the obtained values.

$$\left(s-12\right)\left(s-12\right)$$

Rewrite as a binomial square.

$$\left(s-12\right)^{2}$$

To find equation solution, solve s-12=0.

$$s=12$$

Steps Using Factoring By Grouping

Subtract 25 from both sides.

$$s^{2}-24s+169-25=0$$

Subtract 25 from 169 to get 144.

$$s^{2}-24s+144=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as s^{2}+as+bs+144. To find a and b, set up a system to be solved.

$$a+b=-24 ab=1\times 144=144$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Calculate the sum for each pair.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

The solution is the pair that gives sum -24.

$$a=-12 b=-12$$

Rewrite s^{2}-24s+144 as \left(s^{2}-12s\right)+\left(-12s+144\right).

$$\left(s^{2}-12s\right)+\left(-12s+144\right)$$

Factor out s in the first and -12 in the second group.

$$s\left(s-12\right)-12\left(s-12\right)$$

Factor out common term s-12 by using distributive property.

$$\left(s-12\right)\left(s-12\right)$$

Rewrite as a binomial square.

$$\left(s-12\right)^{2}$$

To find equation solution, solve s-12=0.

$$s=12$$

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.

$$s^{2}-24s+169=25$$

Subtract 25 from both sides of the equation.

$$s^{2}-24s+169-25=25-25$$

Subtracting 25 from itself leaves 0.

$$s^{2}-24s+169-25=0$$

Subtract 25 from 169.

$$s^{2}-24s+144=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -24 for b, and 144 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 144}}{2}$$

Square -24.

$$s=\frac{-\left(-24\right)±\sqrt{576-4\times 144}}{2}$$

Multiply -4 times 144.

$$s=\frac{-\left(-24\right)±\sqrt{576-576}}{2}$$

Add 576 to -576.

$$s=\frac{-\left(-24\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$s=-\frac{-24}{2}$$

The opposite of -24 is 24.

$$s=\frac{24}{2}$$

Divide 24 by 2.

$$s=12$$

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.

$$s^{2}-24s+169=25$$

Subtract 169 from both sides of the equation.

$$s^{2}-24s+169-169=25-169$$

Subtracting 169 from itself leaves 0.

$$s^{2}-24s=25-169$$

Subtract 169 from 25.

$$s^{2}-24s=-144$$

Divide -24, the coefficient of the x term, by 2 to get -12. Then add the square of -12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$s^{2}-24s+\left(-12\right)^{2}=-144+\left(-12\right)^{2}$$

Square -12.

$$s^{2}-24s+144=-144+144$$

Add -144 to 144.

$$s^{2}-24s+144=0$$

Factor s^{2}-24s+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(s-12\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(s-12\right)^{2}}=\sqrt{0}$$

Simplify.

$$s-12=0 s-12=0$$

Add 12 to both sides of the equation.

$$s=12 s=12$$

The equation is now solved. Solutions are the same.

$$s=12$$