Solve 4x^2-11x-3

$$4 { x }^{ 2 } -11x-3$$

Factor

$$\left(x-3\right)\left(4x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-11 ab=4\left(-3\right)=-12$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -12.

$$1,-12 2,-6 3,-4$$

Calculate the sum for each pair.

$$1-12=-11 2-6=-4 3-4=-1$$

The solution is the pair that gives sum -11.

$$a=-12 b=1$$

Rewrite 4x^{2}-11x-3 as \left(4x^{2}-12x\right)+\left(x-3\right).

$$\left(4x^{2}-12x\right)+\left(x-3\right)$$

Factor out 4x in 4x^{2}-12x.

$$4x\left(x-3\right)+x-3$$

Factor out common term x-3 by using distributive property.

$$\left(x-3\right)\left(4x+1\right)$$

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.

$$4x^{2}-11x-3=0$$

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.

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

Square -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 4\left(-3\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-11\right)±\sqrt{121-16\left(-3\right)}}{2\times 4}$$

Multiply -16 times -3.

$$x=\frac{-\left(-11\right)±\sqrt{121+48}}{2\times 4}$$

Add 121 to 48.

$$x=\frac{-\left(-11\right)±\sqrt{169}}{2\times 4}$$

Take the square root of 169.

$$x=\frac{-\left(-11\right)±13}{2\times 4}$$

The opposite of -11 is 11.

$$x=\frac{11±13}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{11±13}{8}$$

Now solve the equation x=\frac{11±13}{8} when ± is plus. Add 11 to 13.

$$x=\frac{24}{8}$$

Divide 24 by 8.

$$x=3$$

Now solve the equation x=\frac{11±13}{8} when ± is minus. Subtract 13 from 11.

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

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 3 for x_{1} and -\frac{1}{4} for x_{2}.

$$4x^{2}-11x-3=4\left(x-3\right)\left(x-\left(-\frac{1}{4}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$4x^{2}-11x-3=4\left(x-3\right)\left(x+\frac{1}{4}\right)$$

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

$$4x^{2}-11x-3=4\left(x-3\right)\times \frac{4x+1}{4}$$

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

$$4x^{2}-11x-3=\left(x-3\right)\left(4x+1\right)$$

Evaluate

$$\left(x-3\right)\left(4x+1\right)$$