Solve 4z^2+23z+15

$$4 { z }^{ 2 } +23z+15$$

Factor

$$\left(z+5\right)\left(4z+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 4z^{2}+az+bz+15. To find a and b, set up a system to be solved.

$$a+b=23 ab=4\times 15=60$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 60.

$$1,60 2,30 3,20 4,15 5,12 6,10$$

Calculate the sum for each pair.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

The solution is the pair that gives sum 23.

$$a=3 b=20$$

Rewrite 4z^{2}+23z+15 as \left(4z^{2}+3z\right)+\left(20z+15\right).

$$\left(4z^{2}+3z\right)+\left(20z+15\right)$$

Factor out z in the first and 5 in the second group.

$$z\left(4z+3\right)+5\left(4z+3\right)$$

Factor out common term 4z+3 by using distributive property.

$$\left(4z+3\right)\left(z+5\right)$$

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.

$$4z^{2}+23z+15=0$$

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.

$$z=\frac{-23±\sqrt{23^{2}-4\times 4\times 15}}{2\times 4}$$

Square 23.

$$z=\frac{-23±\sqrt{529-4\times 4\times 15}}{2\times 4}$$

Multiply -4 times 4.

$$z=\frac{-23±\sqrt{529-16\times 15}}{2\times 4}$$

Multiply -16 times 15.

$$z=\frac{-23±\sqrt{529-240}}{2\times 4}$$

Add 529 to -240.

$$z=\frac{-23±\sqrt{289}}{2\times 4}$$

Take the square root of 289.

$$z=\frac{-23±17}{2\times 4}$$

Multiply 2 times 4.

$$z=\frac{-23±17}{8}$$

Now solve the equation z=\frac{-23±17}{8} when ± is plus. Add -23 to 17.

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

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

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

Now solve the equation z=\frac{-23±17}{8} when ± is minus. Subtract 17 from -23.

$$z=-\frac{40}{8}$$

Divide -40 by 8.

$$z=-5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{4} for x_{1} and -5 for x_{2}.

$$4z^{2}+23z+15=4\left(z-\left(-\frac{3}{4}\right)\right)\left(z-\left(-5\right)\right)$$

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

$$4z^{2}+23z+15=4\left(z+\frac{3}{4}\right)\left(z+5\right)$$

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

$$4z^{2}+23z+15=4\times \frac{4z+3}{4}\left(z+5\right)$$

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

$$4z^{2}+23z+15=\left(4z+3\right)\left(z+5\right)$$

Evaluate

$$\left(z+5\right)\left(4z+3\right)$$