Solve 6z^2-23z+7

$$6 z ^ { 2 } - 23 z + 7$$

Factor

$$\left(2z-7\right)\left(3z-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6z^{2}+az+bz+7. To find a and b, set up a system to be solved.

$$a+b=-23 ab=6\times 7=42$$

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 42.

$$-1,-42 -2,-21 -3,-14 -6,-7$$

Calculate the sum for each pair.

$$-1-42=-43 -2-21=-23 -3-14=-17 -6-7=-13$$

The solution is the pair that gives sum -23.

$$a=-21 b=-2$$

Rewrite 6z^{2}-23z+7 as \left(6z^{2}-21z\right)+\left(-2z+7\right).

$$\left(6z^{2}-21z\right)+\left(-2z+7\right)$$

Factor out 3z in the first and -1 in the second group.

$$3z\left(2z-7\right)-\left(2z-7\right)$$

Factor out common term 2z-7 by using distributive property.

$$\left(2z-7\right)\left(3z-1\right)$$

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.

$$6z^{2}-23z+7=0$$

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.

$$z=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 6\times 7}}{2\times 6}$$

Square -23.

$$z=\frac{-\left(-23\right)±\sqrt{529-4\times 6\times 7}}{2\times 6}$$

Multiply -4 times 6.

$$z=\frac{-\left(-23\right)±\sqrt{529-24\times 7}}{2\times 6}$$

Multiply -24 times 7.

$$z=\frac{-\left(-23\right)±\sqrt{529-168}}{2\times 6}$$

Add 529 to -168.

$$z=\frac{-\left(-23\right)±\sqrt{361}}{2\times 6}$$

Take the square root of 361.

$$z=\frac{-\left(-23\right)±19}{2\times 6}$$

The opposite of -23 is 23.

$$z=\frac{23±19}{2\times 6}$$

Multiply 2 times 6.

$$z=\frac{23±19}{12}$$

Now solve the equation z=\frac{23±19}{12} when ± is plus. Add 23 to 19.

$$z=\frac{42}{12}$$

Reduce the fraction \frac{42}{12} to lowest terms by extracting and canceling out 6.

$$z=\frac{7}{2}$$

Now solve the equation z=\frac{23±19}{12} when ± is minus. Subtract 19 from 23.

$$z=\frac{4}{12}$$

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

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{2} for x_{1} and \frac{1}{3} for x_{2}.

$$6z^{2}-23z+7=6\left(z-\frac{7}{2}\right)\left(z-\frac{1}{3}\right)$$

Subtract \frac{7}{2} from z by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$6z^{2}-23z+7=6\times \frac{2z-7}{2}\left(z-\frac{1}{3}\right)$$

Subtract \frac{1}{3} from z by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$6z^{2}-23z+7=6\times \frac{2z-7}{2}\times \frac{3z-1}{3}$$

Multiply \frac{2z-7}{2} times \frac{3z-1}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$6z^{2}-23z+7=6\times \frac{\left(2z-7\right)\left(3z-1\right)}{2\times 3}$$

Multiply 2 times 3.

$$6z^{2}-23z+7=6\times \frac{\left(2z-7\right)\left(3z-1\right)}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6z^{2}-23z+7=\left(2z-7\right)\left(3z-1\right)$$

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 6

$$x ^ 2 -\frac{23}{6}x +\frac{7}{6} = 0$$

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

$$r + s = \frac{23}{6} rs = \frac{7}{6}$$

Two numbers r and s sum up to \frac{23}{6} exactly when the average of the two numbers is \frac{1}{2}*\frac{23}{6} = \frac{23}{12}. 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>

$$r = \frac{23}{12} - u s = \frac{23}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{7}{6}

$$(\frac{23}{12} - u) (\frac{23}{12} + u) = \frac{7}{6}$$

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

$$\frac{529}{144} - u^2 = \frac{7}{6}$$

Simplify the expression by subtracting \frac{529}{144} on both sides

$$-u^2 = \frac{7}{6}-\frac{529}{144} = -\frac{361}{144}$$

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

$$u^2 = \frac{361}{144} u = \pm\sqrt{\frac{361}{144}} = \pm \frac{19}{12}$$

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

$$r =\frac{23}{12} - \frac{19}{12} = 0.333 s = \frac{23}{12} + \frac{19}{12} = 3.500$$

Evaluate

$$\left(2z-7\right)\left(3z-1\right)$$