Solve 4z^2-16z+15=0

$$4 z ^ { 2 } - 16 z + 15 = 0$$

Solve for z

$$z = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$z = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4z^{2}+az+bz+15. To find a and b, set up a system to be solved.

$$a+b=-16 ab=4\times 15=60$$

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 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 -16.

$$a=-10 b=-6$$

Rewrite 4z^{2}-16z+15 as \left(4z^{2}-10z\right)+\left(-6z+15\right).

$$\left(4z^{2}-10z\right)+\left(-6z+15\right)$$

Factor out 2z in the first and -3 in the second group.

$$2z\left(2z-5\right)-3\left(2z-5\right)$$

Factor out common term 2z-5 by using distributive property.

$$\left(2z-5\right)\left(2z-3\right)$$

To find equation solutions, solve 2z-5=0 and 2z-3=0.

$$z=\frac{5}{2} z=\frac{3}{2}$$

Steps Using the Quadratic Formula

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.

$$4z^{2}-16z+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -16 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 4\times 15}}{2\times 4}$$

Square -16.

$$z=\frac{-\left(-16\right)±\sqrt{256-4\times 4\times 15}}{2\times 4}$$

Multiply -4 times 4.

$$z=\frac{-\left(-16\right)±\sqrt{256-16\times 15}}{2\times 4}$$

Multiply -16 times 15.

$$z=\frac{-\left(-16\right)±\sqrt{256-240}}{2\times 4}$$

Add 256 to -240.

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

Take the square root of 16.

$$z=\frac{-\left(-16\right)±4}{2\times 4}$$

The opposite of -16 is 16.

$$z=\frac{16±4}{2\times 4}$$

Multiply 2 times 4.

$$z=\frac{16±4}{8}$$

Now solve the equation z=\frac{16±4}{8} when ± is plus. Add 16 to 4.

$$z=\frac{20}{8}$$

Reduce the fraction \frac{20}{8} to lowest terms by extracting and canceling out 4.

$$z=\frac{5}{2}$$

Now solve the equation z=\frac{16±4}{8} when ± is minus. Subtract 4 from 16.

$$z=\frac{12}{8}$$

Reduce the fraction \frac{12}{8} to lowest terms by extracting and canceling out 4.

$$z=\frac{3}{2}$$

The equation is now solved.

$$z=\frac{5}{2} z=\frac{3}{2}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$4z^{2}-16z+15=0$$

Subtract 15 from both sides of the equation.

$$4z^{2}-16z+15-15=-15$$

Subtracting 15 from itself leaves 0.

$$4z^{2}-16z=-15$$

Divide both sides by 4.

$$\frac{4z^{2}-16z}{4}=-\frac{15}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$z^{2}+\left(-\frac{16}{4}\right)z=-\frac{15}{4}$$

Divide -16 by 4.

$$z^{2}-4z=-\frac{15}{4}$$

Divide -4, the coefficient of the x term, by 2 to get -2. Then add the square of -2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Square -2.

$$z^{2}-4z+4=-\frac{15}{4}+4$$

Add -\frac{15}{4} to 4.

$$z^{2}-4z+4=\frac{1}{4}$$

Factor z^{2}-4z+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(z-2\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(z-2\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$z-2=\frac{1}{2} z-2=-\frac{1}{2}$$

Add 2 to both sides of the equation.

$$z=\frac{5}{2} z=\frac{3}{2}$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.This is achieved by dividing both sides of the equation by 4

$$x ^ 2 -4x +\frac{15}{4} = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 4 rs = \frac{15}{4}$$

Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 2. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = 2 - u s = 2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{15}{4}

$$(2 - u) (2 + u) = \frac{15}{4}$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$4 - u^2 = \frac{15}{4}$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = \frac{15}{4}-4 = -\frac{1}{4}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =2 - \frac{1}{2} = 1.500 s = 2 + \frac{1}{2} = 2.500$$