Solve x^2-138x+4761

$${ x }^{ 2 } -138x+4761$$

Factor

$$\left(x-69\right)^{2}$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+4761. To find a and b, set up a system to be solved.

$$a+b=-138 ab=1\times 4761=4761$$

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 4761.

$$-1,-4761 -3,-1587 -9,-529 -23,-207 -69,-69$$

Calculate the sum for each pair.

$$-1-4761=-4762 -3-1587=-1590 -9-529=-538 -23-207=-230 -69-69=-138$$

The solution is the pair that gives sum -138.

$$a=-69 b=-69$$

Rewrite x^{2}-138x+4761 as \left(x^{2}-69x\right)+\left(-69x+4761\right).

$$\left(x^{2}-69x\right)+\left(-69x+4761\right)$$

Factor out x in the first and -69 in the second group.

$$x\left(x-69\right)-69\left(x-69\right)$$

Factor out common term x-69 by using distributive property.

$$\left(x-69\right)\left(x-69\right)$$

Rewrite as a binomial square.

$$\left(x-69\right)^{2}$$

Steps Using Square Of Binomial

This trinomial has the form of a trinomial square, perhaps multiplied by a common factor. Trinomial squares can be factored by finding the square roots of the leading and trailing terms.

$$factor(x^{2}-138x+4761)$$

Find the square root of the trailing term, 4761.

$$\sqrt{4761}=69$$

The trinomial square is the square of the binomial that is the sum or difference of the square roots of the leading and trailing terms, with the sign determined by the sign of the middle term of the trinomial square.

$$\left(x-69\right)^{2}$$

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.

$$x^{2}-138x+4761=0$$

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.

$$x=\frac{-\left(-138\right)±\sqrt{\left(-138\right)^{2}-4\times 4761}}{2}$$

Square -138.

$$x=\frac{-\left(-138\right)±\sqrt{19044-4\times 4761}}{2}$$

Multiply -4 times 4761.

$$x=\frac{-\left(-138\right)±\sqrt{19044-19044}}{2}$$

Add 19044 to -19044.

$$x=\frac{-\left(-138\right)±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=\frac{-\left(-138\right)±0}{2}$$

The opposite of -138 is 138.

$$x=\frac{138±0}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 69 for x_{1} and 69 for x_{2}.

$$x^{2}-138x+4761=\left(x-69\right)\left(x-69\right)$$

Evaluate

$$\left(x-69\right)^{2}$$