Solve 6b^2-43b+42

$$6 b ^ { 2 } - 43 b + 42$$

Factor

$$\left(b-6\right)\left(6b-7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6b^{2}+pb+qb+42. To find p and q, set up a system to be solved.

$$p+q=-43 pq=6\times 42=252$$

Since pq is positive, p and q have the same sign. Since p+q is negative, p and q are both negative. List all such integer pairs that give product 252.

$$-1,-252 -2,-126 -3,-84 -4,-63 -6,-42 -7,-36 -9,-28 -12,-21 -14,-18$$

Calculate the sum for each pair.

$$-1-252=-253 -2-126=-128 -3-84=-87 -4-63=-67 -6-42=-48 -7-36=-43 -9-28=-37 -12-21=-33 -14-18=-32$$

The solution is the pair that gives sum -43.

$$p=-36 q=-7$$

Rewrite 6b^{2}-43b+42 as \left(6b^{2}-36b\right)+\left(-7b+42\right).

$$\left(6b^{2}-36b\right)+\left(-7b+42\right)$$

Factor out 6b in the first and -7 in the second group.

$$6b\left(b-6\right)-7\left(b-6\right)$$

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

$$\left(b-6\right)\left(6b-7\right)$$

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.

$$6b^{2}-43b+42=0$$

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.

$$b=\frac{-\left(-43\right)±\sqrt{\left(-43\right)^{2}-4\times 6\times 42}}{2\times 6}$$

Square -43.

$$b=\frac{-\left(-43\right)±\sqrt{1849-4\times 6\times 42}}{2\times 6}$$

Multiply -4 times 6.

$$b=\frac{-\left(-43\right)±\sqrt{1849-24\times 42}}{2\times 6}$$

Multiply -24 times 42.

$$b=\frac{-\left(-43\right)±\sqrt{1849-1008}}{2\times 6}$$

Add 1849 to -1008.

$$b=\frac{-\left(-43\right)±\sqrt{841}}{2\times 6}$$

Take the square root of 841.

$$b=\frac{-\left(-43\right)±29}{2\times 6}$$

The opposite of -43 is 43.

$$b=\frac{43±29}{2\times 6}$$

Multiply 2 times 6.

$$b=\frac{43±29}{12}$$

Now solve the equation b=\frac{43±29}{12} when ± is plus. Add 43 to 29.

$$b=\frac{72}{12}$$

Divide 72 by 12.

$$b=6$$

Now solve the equation b=\frac{43±29}{12} when ± is minus. Subtract 29 from 43.

$$b=\frac{14}{12}$$

Reduce the fraction \frac{14}{12} to lowest terms by extracting and canceling out 2.

$$b=\frac{7}{6}$$

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{7}{6} for x_{2}.

$$6b^{2}-43b+42=6\left(b-6\right)\left(b-\frac{7}{6}\right)$$

Subtract \frac{7}{6} from b by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$6b^{2}-43b+42=6\left(b-6\right)\times \frac{6b-7}{6}$$

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

$$6b^{2}-43b+42=\left(b-6\right)\left(6b-7\right)$$

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

$$x ^ 2 -\frac{43}{6}x +7 = 0$$

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

$$r + s = \frac{43}{6} rs = 7$$

Two numbers r and s sum up to \frac{43}{6} exactly when the average of the two numbers is \frac{1}{2}*\frac{43}{6} = \frac{43}{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>

$$r = \frac{43}{12} - u s = \frac{43}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 7

$$(\frac{43}{12} - u) (\frac{43}{12} + u) = 7$$

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

$$\frac{1849}{144} - u^2 = 7$$

Simplify the expression by subtracting \frac{1849}{144} on both sides

$$-u^2 = 7-\frac{1849}{144} = -\frac{841}{144}$$

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

$$u^2 = \frac{841}{144} u = \pm\sqrt{\frac{841}{144}} = \pm \frac{29}{12}$$

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

$$r =\frac{43}{12} - \frac{29}{12} = 1.167 s = \frac{43}{12} + \frac{29}{12} = 6$$

Evaluate

$$\left(b-6\right)\left(6b-7\right)$$