Solve 4x^2-25x+36

$$4 { x }^{ 2 } -25x+36$$

Factor

$$\left(x-4\right)\left(4x-9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+36. To find a and b, set up a system to be solved.

$$a+b=-25 ab=4\times 36=144$$

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 -25.

$$a=-16 b=-9$$

Rewrite 4x^{2}-25x+36 as \left(4x^{2}-16x\right)+\left(-9x+36\right).

$$\left(4x^{2}-16x\right)+\left(-9x+36\right)$$

Factor out 4x in the first and -9 in the second group.

$$4x\left(x-4\right)-9\left(x-4\right)$$

Factor out common term x-4 by using distributive property.

$$\left(x-4\right)\left(4x-9\right)$$

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.

$$4x^{2}-25x+36=0$$

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.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 4\times 36}}{2\times 4}$$

Square -25.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 4\times 36}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-25\right)±\sqrt{625-16\times 36}}{2\times 4}$$

Multiply -16 times 36.

$$x=\frac{-\left(-25\right)±\sqrt{625-576}}{2\times 4}$$

Add 625 to -576.

$$x=\frac{-\left(-25\right)±\sqrt{49}}{2\times 4}$$

Take the square root of 49.

$$x=\frac{-\left(-25\right)±7}{2\times 4}$$

The opposite of -25 is 25.

$$x=\frac{25±7}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{25±7}{8}$$

Now solve the equation x=\frac{25±7}{8} when ± is plus. Add 25 to 7.

$$x=\frac{32}{8}$$

Divide 32 by 8.

$$x=4$$

Now solve the equation x=\frac{25±7}{8} when ± is minus. Subtract 7 from 25.

$$x=\frac{18}{8}$$

Reduce the fraction \frac{18}{8} to lowest terms by extracting and canceling out 2.

$$x=\frac{9}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 4 for x_{1} and \frac{9}{4} for x_{2}.

$$4x^{2}-25x+36=4\left(x-4\right)\left(x-\frac{9}{4}\right)$$

Subtract \frac{9}{4} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-25x+36=4\left(x-4\right)\times \frac{4x-9}{4}$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4x^{2}-25x+36=\left(x-4\right)\left(4x-9\right)$$

Evaluate

$$\left(x-4\right)\left(4x-9\right)$$