Solve 52z^2-43z+3

$$52 z ^ { 2 } - 43 z + 3$$

Factor

$$\left(4z-3\right)\left(13z-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 52z^{2}+az+bz+3. To find a and b, set up a system to be solved.

$$a+b=-43 ab=52\times 3=156$$

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 156.

$$-1,-156 -2,-78 -3,-52 -4,-39 -6,-26 -12,-13$$

Calculate the sum for each pair.

$$-1-156=-157 -2-78=-80 -3-52=-55 -4-39=-43 -6-26=-32 -12-13=-25$$

The solution is the pair that gives sum -43.

$$a=-39 b=-4$$

Rewrite 52z^{2}-43z+3 as \left(52z^{2}-39z\right)+\left(-4z+3\right).

$$\left(52z^{2}-39z\right)+\left(-4z+3\right)$$

Factor out 13z in the first and -1 in the second group.

$$13z\left(4z-3\right)-\left(4z-3\right)$$

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

$$\left(4z-3\right)\left(13z-1\right)$$

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.

$$52z^{2}-43z+3=0$$

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.

$$z=\frac{-\left(-43\right)±\sqrt{\left(-43\right)^{2}-4\times 52\times 3}}{2\times 52}$$

Square -43.

$$z=\frac{-\left(-43\right)±\sqrt{1849-4\times 52\times 3}}{2\times 52}$$

Multiply -4 times 52.

$$z=\frac{-\left(-43\right)±\sqrt{1849-208\times 3}}{2\times 52}$$

Multiply -208 times 3.

$$z=\frac{-\left(-43\right)±\sqrt{1849-624}}{2\times 52}$$

Add 1849 to -624.

$$z=\frac{-\left(-43\right)±\sqrt{1225}}{2\times 52}$$

Take the square root of 1225.

$$z=\frac{-\left(-43\right)±35}{2\times 52}$$

The opposite of -43 is 43.

$$z=\frac{43±35}{2\times 52}$$

Multiply 2 times 52.

$$z=\frac{43±35}{104}$$

Now solve the equation z=\frac{43±35}{104} when ± is plus. Add 43 to 35.

$$z=\frac{78}{104}$$

Reduce the fraction \frac{78}{104} to lowest terms by extracting and canceling out 26.

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

Now solve the equation z=\frac{43±35}{104} when ± is minus. Subtract 35 from 43.

$$z=\frac{8}{104}$$

Reduce the fraction \frac{8}{104} to lowest terms by extracting and canceling out 8.

$$z=\frac{1}{13}$$

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 \frac{1}{13} for x_{2}.

$$52z^{2}-43z+3=52\left(z-\frac{3}{4}\right)\left(z-\frac{1}{13}\right)$$

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

$$52z^{2}-43z+3=52\times \frac{4z-3}{4}\left(z-\frac{1}{13}\right)$$

Subtract \frac{1}{13} from z by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$52z^{2}-43z+3=52\times \frac{4z-3}{4}\times \frac{13z-1}{13}$$

Multiply \frac{4z-3}{4} times \frac{13z-1}{13} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$52z^{2}-43z+3=52\times \frac{\left(4z-3\right)\left(13z-1\right)}{4\times 13}$$

Multiply 4 times 13.

$$52z^{2}-43z+3=52\times \frac{\left(4z-3\right)\left(13z-1\right)}{52}$$

Cancel out 52, the greatest common factor in 52 and 52.

$$52z^{2}-43z+3=\left(4z-3\right)\left(13z-1\right)$$

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 52

$$x ^ 2 -\frac{43}{52}x +\frac{3}{52} = 0$$

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

$$r + s = \frac{43}{52} rs = \frac{3}{52}$$

Two numbers r and s sum up to \frac{43}{52} exactly when the average of the two numbers is \frac{1}{2}*\frac{43}{52} = \frac{43}{104}. 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>

$$r = \frac{43}{104} - u s = \frac{43}{104} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{52}

$$(\frac{43}{104} - u) (\frac{43}{104} + u) = \frac{3}{52}$$

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

$$\frac{1849}{10816} - u^2 = \frac{3}{52}$$

Simplify the expression by subtracting \frac{1849}{10816} on both sides

$$-u^2 = \frac{3}{52}-\frac{1849}{10816} = -\frac{1225}{10816}$$

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

$$u^2 = \frac{1225}{10816} u = \pm\sqrt{\frac{1225}{10816}} = \pm \frac{35}{104}$$

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

$$r =\frac{43}{104} - \frac{35}{104} = 0.07692307692307693 s = \frac{43}{104} + \frac{35}{104} = 0.75$$

Evaluate

$$\left(4z-3\right)\left(13z-1\right)$$