Solve 4x^2-17x+4

$$4 { x }^{ 2 } -17x+4$$

Factor

$$\left(x-4\right)\left(4x-1\right)$$

Solution Steps

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

$$a+b=-17 ab=4\times 4=16$$

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 16.

$$-1,-16 -2,-8 -4,-4$$

Calculate the sum for each pair.

$$-1-16=-17 -2-8=-10 -4-4=-8$$

The solution is the pair that gives sum -17.

$$a=-16 b=-1$$

Rewrite 4x^{2}-17x+4 as \left(4x^{2}-16x\right)+\left(-x+4\right).

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

Factor out 4x in the first and -1 in the second group.

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

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

$$\left(x-4\right)\left(4x-1\right)$$

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.

$$4x^{2}-17x+4=0$$

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.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 4\times 4}}{2\times 4}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 4\times 4}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-17\right)±\sqrt{289-16\times 4}}{2\times 4}$$

Multiply -16 times 4.

$$x=\frac{-\left(-17\right)±\sqrt{289-64}}{2\times 4}$$

Add 289 to -64.

$$x=\frac{-\left(-17\right)±\sqrt{225}}{2\times 4}$$

Take the square root of 225.

$$x=\frac{-\left(-17\right)±15}{2\times 4}$$

The opposite of -17 is 17.

$$x=\frac{17±15}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{17±15}{8}$$

Now solve the equation x=\frac{17±15}{8} when ± is plus. Add 17 to 15.

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

Divide 32 by 8.

$$x=4$$

Now solve the equation x=\frac{17±15}{8} when ± is minus. Subtract 15 from 17.

$$x=\frac{2}{8}$$

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

$$x=\frac{1}{4}$$

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{1}{4} for x_{2}.

$$4x^{2}-17x+4=4\left(x-4\right)\left(x-\frac{1}{4}\right)$$

Subtract \frac{1}{4} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-17x+4=4\left(x-4\right)\times \frac{4x-1}{4}$$

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

$$4x^{2}-17x+4=\left(x-4\right)\left(4x-1\right)$$

Evaluate

$$\left(x-4\right)\left(4x-1\right)$$