Solve 4x^2-21x-18

$$4 { x }^{ 2 } -21x-18$$

Factor

$$\left(x-6\right)\left(4x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=-21 ab=4\left(-18\right)=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Calculate the sum for each pair.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

The solution is the pair that gives sum -21.

$$a=-24 b=3$$

Rewrite 4x^{2}-21x-18 as \left(4x^{2}-24x\right)+\left(3x-18\right).

$$\left(4x^{2}-24x\right)+\left(3x-18\right)$$

Factor out 4x in the first and 3 in the second group.

$$4x\left(x-6\right)+3\left(x-6\right)$$

Factor out common term x-6 by using distributive property.

$$\left(x-6\right)\left(4x+3\right)$$

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.

$$4x^{2}-21x-18=0$$

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.

$$x=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 4\left(-18\right)}}{2\times 4}$$

Square -21.

$$x=\frac{-\left(-21\right)±\sqrt{441-4\times 4\left(-18\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-21\right)±\sqrt{441-16\left(-18\right)}}{2\times 4}$$

Multiply -16 times -18.

$$x=\frac{-\left(-21\right)±\sqrt{441+288}}{2\times 4}$$

Add 441 to 288.

$$x=\frac{-\left(-21\right)±\sqrt{729}}{2\times 4}$$

Take the square root of 729.

$$x=\frac{-\left(-21\right)±27}{2\times 4}$$

The opposite of -21 is 21.

$$x=\frac{21±27}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{21±27}{8}$$

Now solve the equation x=\frac{21±27}{8} when ± is plus. Add 21 to 27.

$$x=\frac{48}{8}$$

Divide 48 by 8.

$$x=6$$

Now solve the equation x=\frac{21±27}{8} when ± is minus. Subtract 27 from 21.

$$x=-\frac{6}{8}$$

Reduce the fraction \frac{-6}{8} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and -\frac{3}{4} for x_{2}.

$$4x^{2}-21x-18=4\left(x-6\right)\left(x-\left(-\frac{3}{4}\right)\right)$$

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

$$4x^{2}-21x-18=4\left(x-6\right)\left(x+\frac{3}{4}\right)$$

Add \frac{3}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4x^{2}-21x-18=4\left(x-6\right)\times \frac{4x+3}{4}$$

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

$$4x^{2}-21x-18=\left(x-6\right)\left(4x+3\right)$$

Evaluate

$$\left(x-6\right)\left(4x+3\right)$$