Solve 7x^2-40x-12=

$$7 x ^ { 2 } - 40 x - 12 =$$

Factor

$$\left(x-6\right)\left(7x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 7x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=-40 ab=7\left(-12\right)=-84$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -84.

$$1,-84 2,-42 3,-28 4,-21 6,-14 7,-12$$

Calculate the sum for each pair.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

The solution is the pair that gives sum -40.

$$a=-42 b=2$$

Rewrite 7x^{2}-40x-12 as \left(7x^{2}-42x\right)+\left(2x-12\right).

$$\left(7x^{2}-42x\right)+\left(2x-12\right)$$

Factor out 7x in the first and 2 in the second group.

$$7x\left(x-6\right)+2\left(x-6\right)$$

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

$$\left(x-6\right)\left(7x+2\right)$$

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.

$$7x^{2}-40x-12=0$$

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.

$$x=\frac{-\left(-40\right)±\sqrt{\left(-40\right)^{2}-4\times 7\left(-12\right)}}{2\times 7}$$

Square -40.

$$x=\frac{-\left(-40\right)±\sqrt{1600-4\times 7\left(-12\right)}}{2\times 7}$$

Multiply -4 times 7.

$$x=\frac{-\left(-40\right)±\sqrt{1600-28\left(-12\right)}}{2\times 7}$$

Multiply -28 times -12.

$$x=\frac{-\left(-40\right)±\sqrt{1600+336}}{2\times 7}$$

Add 1600 to 336.

$$x=\frac{-\left(-40\right)±\sqrt{1936}}{2\times 7}$$

Take the square root of 1936.

$$x=\frac{-\left(-40\right)±44}{2\times 7}$$

The opposite of -40 is 40.

$$x=\frac{40±44}{2\times 7}$$

Multiply 2 times 7.

$$x=\frac{40±44}{14}$$

Now solve the equation x=\frac{40±44}{14} when ± is plus. Add 40 to 44.

$$x=\frac{84}{14}$$

Divide 84 by 14.

$$x=6$$

Now solve the equation x=\frac{40±44}{14} when ± is minus. Subtract 44 from 40.

$$x=-\frac{4}{14}$$

Reduce the fraction \frac{-4}{14} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{7}$$

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

$$7x^{2}-40x-12=7\left(x-6\right)\left(x-\left(-\frac{2}{7}\right)\right)$$

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

$$7x^{2}-40x-12=7\left(x-6\right)\left(x+\frac{2}{7}\right)$$

Add \frac{2}{7} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$7x^{2}-40x-12=7\left(x-6\right)\times \frac{7x+2}{7}$$

Cancel out 7, the greatest common factor in 7 and 7.

$$7x^{2}-40x-12=\left(x-6\right)\left(7x+2\right)$$

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 7

$$x ^ 2 -\frac{40}{7}x -\frac{12}{7} = 0$$

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

$$r + s = \frac{40}{7} rs = -\frac{12}{7}$$

Two numbers r and s sum up to \frac{40}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{40}{7} = \frac{20}{7}. 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>

$$r = \frac{20}{7} - u s = \frac{20}{7} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{12}{7}

$$(\frac{20}{7} - u) (\frac{20}{7} + u) = -\frac{12}{7}$$

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

$$\frac{400}{49} - u^2 = -\frac{12}{7}$$

Simplify the expression by subtracting \frac{400}{49} on both sides

$$-u^2 = -\frac{12}{7}-\frac{400}{49} = -\frac{484}{49}$$

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

$$u^2 = \frac{484}{49} u = \pm\sqrt{\frac{484}{49}} = \pm \frac{22}{7}$$

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

$$r =\frac{20}{7} - \frac{22}{7} = -0.286 s = \frac{20}{7} + \frac{22}{7} = 6$$

Evaluate

$$\left(x-6\right)\left(7x+2\right)$$