Solve 4x^2-11x+7

$$4 { x }^{ 2 } -11x+7$$

Factor

$$\left(x-1\right)\left(4x-7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+7. To find a and b, set up a system to be solved.

$$a+b=-11 ab=4\times 7=28$$

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 28.

$$-1,-28 -2,-14 -4,-7$$

Calculate the sum for each pair.

$$-1-28=-29 -2-14=-16 -4-7=-11$$

The solution is the pair that gives sum -11.

$$a=-7 b=-4$$

Rewrite 4x^{2}-11x+7 as \left(4x^{2}-7x\right)+\left(-4x+7\right).

$$\left(4x^{2}-7x\right)+\left(-4x+7\right)$$

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

$$x\left(4x-7\right)-\left(4x-7\right)$$

Factor out common term 4x-7 by using distributive property.

$$\left(4x-7\right)\left(x-1\right)$$

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.

$$4x^{2}-11x+7=0$$

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.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 4\times 7}}{2\times 4}$$

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 4\times 7}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-11\right)±\sqrt{121-16\times 7}}{2\times 4}$$

Multiply -16 times 7.

$$x=\frac{-\left(-11\right)±\sqrt{121-112}}{2\times 4}$$

Add 121 to -112.

$$x=\frac{-\left(-11\right)±\sqrt{9}}{2\times 4}$$

Take the square root of 9.

$$x=\frac{-\left(-11\right)±3}{2\times 4}$$

The opposite of -11 is 11.

$$x=\frac{11±3}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{11±3}{8}$$

Now solve the equation x=\frac{11±3}{8} when ± is plus. Add 11 to 3.

$$x=\frac{14}{8}$$

Reduce the fraction \frac{14}{8} to lowest terms by extracting and canceling out 2.

$$x=\frac{7}{4}$$

Now solve the equation x=\frac{11±3}{8} when ± is minus. Subtract 3 from 11.

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

Divide 8 by 8.

$$x=1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{4} for x_{1} and 1 for x_{2}.

$$4x^{2}-11x+7=4\left(x-\frac{7}{4}\right)\left(x-1\right)$$

Subtract \frac{7}{4} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-11x+7=4\times \frac{4x-7}{4}\left(x-1\right)$$

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

$$4x^{2}-11x+7=\left(4x-7\right)\left(x-1\right)$$

Evaluate

$$\left(x-1\right)\left(4x-7\right)$$