Solve 4x^2-12x+5

$$4 { x }^{ 2 } -12x+5$$

Factor

$$\left(2x-5\right)\left(2x-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=-12 ab=4\times 5=20$$

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 20.

$$-1,-20 -2,-10 -4,-5$$

Calculate the sum for each pair.

$$-1-20=-21 -2-10=-12 -4-5=-9$$

The solution is the pair that gives sum -12.

$$a=-10 b=-2$$

Rewrite 4x^{2}-12x+5 as \left(4x^{2}-10x\right)+\left(-2x+5\right).

$$\left(4x^{2}-10x\right)+\left(-2x+5\right)$$

Factor out 2x in the first and -1 in the second group.

$$2x\left(2x-5\right)-\left(2x-5\right)$$

Factor out common term 2x-5 by using distributive property.

$$\left(2x-5\right)\left(2x-1\right)$$

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.

$$4x^{2}-12x+5=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\times 5}}{2\times 4}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\times 5}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\times 5}}{2\times 4}$$

Multiply -16 times 5.

$$x=\frac{-\left(-12\right)±\sqrt{144-80}}{2\times 4}$$

Add 144 to -80.

$$x=\frac{-\left(-12\right)±\sqrt{64}}{2\times 4}$$

Take the square root of 64.

$$x=\frac{-\left(-12\right)±8}{2\times 4}$$

The opposite of -12 is 12.

$$x=\frac{12±8}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{12±8}{8}$$

Now solve the equation x=\frac{12±8}{8} when ± is plus. Add 12 to 8.

$$x=\frac{20}{8}$$

Reduce the fraction \frac{20}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{5}{2}$$

Now solve the equation x=\frac{12±8}{8} when ± is minus. Subtract 8 from 12.

$$x=\frac{4}{8}$$

Reduce the fraction \frac{4}{8} to lowest terms by extracting and canceling out 4.

$$x=\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5}{2} for x_{1} and \frac{1}{2} for x_{2}.

$$4x^{2}-12x+5=4\left(x-\frac{5}{2}\right)\left(x-\frac{1}{2}\right)$$

Subtract \frac{5}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-12x+5=4\times \frac{2x-5}{2}\left(x-\frac{1}{2}\right)$$

Subtract \frac{1}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-12x+5=4\times \frac{2x-5}{2}\times \frac{2x-1}{2}$$

Multiply \frac{2x-5}{2} times \frac{2x-1}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-12x+5=4\times \frac{\left(2x-5\right)\left(2x-1\right)}{2\times 2}$$

Multiply 2 times 2.

$$4x^{2}-12x+5=4\times \frac{\left(2x-5\right)\left(2x-1\right)}{4}$$

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

$$4x^{2}-12x+5=\left(2x-5\right)\left(2x-1\right)$$

Evaluate

$$\left(2x-5\right)\left(2x-1\right)$$