Solve x^2-26x+25

$${ x }^{ 2 } -26x+25$$

Factor

$$\left(x-25\right)\left(x-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+25. To find a and b, set up a system to be solved.

$$a+b=-26 ab=1\times 25=25$$

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 25.

$$-1,-25 -5,-5$$

Calculate the sum for each pair.

$$-1-25=-26 -5-5=-10$$

The solution is the pair that gives sum -26.

$$a=-25 b=-1$$

Rewrite x^{2}-26x+25 as \left(x^{2}-25x\right)+\left(-x+25\right).

$$\left(x^{2}-25x\right)+\left(-x+25\right)$$

Factor out x in the first and -1 in the second group.

$$x\left(x-25\right)-\left(x-25\right)$$

Factor out common term x-25 by using distributive property.

$$\left(x-25\right)\left(x-1\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}-26x+25=0$$

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.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 25}}{2}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\times 25}}{2}$$

Multiply -4 times 25.

$$x=\frac{-\left(-26\right)±\sqrt{676-100}}{2}$$

Add 676 to -100.

$$x=\frac{-\left(-26\right)±\sqrt{576}}{2}$$

Take the square root of 576.

$$x=\frac{-\left(-26\right)±24}{2}$$

The opposite of -26 is 26.

$$x=\frac{26±24}{2}$$

Now solve the equation x=\frac{26±24}{2} when ± is plus. Add 26 to 24.

$$x=\frac{50}{2}$$

Divide 50 by 2.

$$x=25$$

Now solve the equation x=\frac{26±24}{2} when ± is minus. Subtract 24 from 26.

$$x=\frac{2}{2}$$

Divide 2 by 2.

$$x=1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 25 for x_{1} and 1 for x_{2}.

$$x^{2}-26x+25=\left(x-25\right)\left(x-1\right)$$

Evaluate

$$\left(x-25\right)\left(x-1\right)$$