Solve 4x^2-184x+2100

$$4 { x }^{ 2 } -184x+2100$$

Factor

$$4\left(x-25\right)\left(x-21\right)$$

Solution Steps

Factor out 4.

$$4\left(x^{2}-46x+525\right)$$

Consider x^{2}-46x+525. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+525. To find a and b, set up a system to be solved.

$$a+b=-46 ab=1\times 525=525$$

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 525.

$$-1,-525 -3,-175 -5,-105 -7,-75 -15,-35 -21,-25$$

Calculate the sum for each pair.

$$-1-525=-526 -3-175=-178 -5-105=-110 -7-75=-82 -15-35=-50 -21-25=-46$$

The solution is the pair that gives sum -46.

$$a=-25 b=-21$$

Rewrite x^{2}-46x+525 as \left(x^{2}-25x\right)+\left(-21x+525\right).

$$\left(x^{2}-25x\right)+\left(-21x+525\right)$$

Factor out x in the first and -21 in the second group.

$$x\left(x-25\right)-21\left(x-25\right)$$

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

$$\left(x-25\right)\left(x-21\right)$$

Rewrite the complete factored expression.

$$4\left(x-25\right)\left(x-21\right)$$

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.

$$4x^{2}-184x+2100=0$$

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.

$$x=\frac{-\left(-184\right)±\sqrt{\left(-184\right)^{2}-4\times 4\times 2100}}{2\times 4}$$

Square -184.

$$x=\frac{-\left(-184\right)±\sqrt{33856-4\times 4\times 2100}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-184\right)±\sqrt{33856-16\times 2100}}{2\times 4}$$

Multiply -16 times 2100.

$$x=\frac{-\left(-184\right)±\sqrt{33856-33600}}{2\times 4}$$

Add 33856 to -33600.

$$x=\frac{-\left(-184\right)±\sqrt{256}}{2\times 4}$$

Take the square root of 256.

$$x=\frac{-\left(-184\right)±16}{2\times 4}$$

The opposite of -184 is 184.

$$x=\frac{184±16}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{184±16}{8}$$

Now solve the equation x=\frac{184±16}{8} when ± is plus. Add 184 to 16.

$$x=\frac{200}{8}$$

Divide 200 by 8.

$$x=25$$

Now solve the equation x=\frac{184±16}{8} when ± is minus. Subtract 16 from 184.

$$x=\frac{168}{8}$$

Divide 168 by 8.

$$x=21$$

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 21 for x_{2}.

$$4x^{2}-184x+2100=4\left(x-25\right)\left(x-21\right)$$

Evaluate

$$4\left(x-25\right)\left(x-21\right)$$