Solve 14x^2+17x-6

$$14 x ^ { 2 } + 17 x - 6$$

Factor

$$\left(7x-2\right)\left(2x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 14x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=17 ab=14\left(-6\right)=-84$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. 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 17.

$$a=-4 b=21$$

Rewrite 14x^{2}+17x-6 as \left(14x^{2}-4x\right)+\left(21x-6\right).

$$\left(14x^{2}-4x\right)+\left(21x-6\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(7x-2\right)+3\left(7x-2\right)$$

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

$$\left(7x-2\right)\left(2x+3\right)$$

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.

$$14x^{2}+17x-6=0$$

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.

$$x=\frac{-17±\sqrt{17^{2}-4\times 14\left(-6\right)}}{2\times 14}$$

Square 17.

$$x=\frac{-17±\sqrt{289-4\times 14\left(-6\right)}}{2\times 14}$$

Multiply -4 times 14.

$$x=\frac{-17±\sqrt{289-56\left(-6\right)}}{2\times 14}$$

Multiply -56 times -6.

$$x=\frac{-17±\sqrt{289+336}}{2\times 14}$$

Add 289 to 336.

$$x=\frac{-17±\sqrt{625}}{2\times 14}$$

Take the square root of 625.

$$x=\frac{-17±25}{2\times 14}$$

Multiply 2 times 14.

$$x=\frac{-17±25}{28}$$

Now solve the equation x=\frac{-17±25}{28} when ± is plus. Add -17 to 25.

$$x=\frac{8}{28}$$

Reduce the fraction \frac{8}{28} to lowest terms by extracting and canceling out 4.

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

Now solve the equation x=\frac{-17±25}{28} when ± is minus. Subtract 25 from -17.

$$x=-\frac{42}{28}$$

Reduce the fraction \frac{-42}{28} to lowest terms by extracting and canceling out 14.

$$x=-\frac{3}{2}$$

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

$$14x^{2}+17x-6=14\left(x-\frac{2}{7}\right)\left(x-\left(-\frac{3}{2}\right)\right)$$

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

$$14x^{2}+17x-6=14\left(x-\frac{2}{7}\right)\left(x+\frac{3}{2}\right)$$

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

$$14x^{2}+17x-6=14\times \frac{7x-2}{7}\left(x+\frac{3}{2}\right)$$

Add \frac{3}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$14x^{2}+17x-6=14\times \frac{7x-2}{7}\times \frac{2x+3}{2}$$

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

$$14x^{2}+17x-6=14\times \frac{\left(7x-2\right)\left(2x+3\right)}{7\times 2}$$

Multiply 7 times 2.

$$14x^{2}+17x-6=14\times \frac{\left(7x-2\right)\left(2x+3\right)}{14}$$

Cancel out 14, the greatest common factor in 14 and 14.

$$14x^{2}+17x-6=\left(7x-2\right)\left(2x+3\right)$$

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 14

$$x ^ 2 +\frac{17}{14}x -\frac{3}{7} = 0$$

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

$$r + s = -\frac{17}{14} rs = -\frac{3}{7}$$

Two numbers r and s sum up to -\frac{17}{14} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{14} = -\frac{17}{28}. 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>

$$r = -\frac{17}{28} - u s = -\frac{17}{28} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{3}{7}

$$(-\frac{17}{28} - u) (-\frac{17}{28} + u) = -\frac{3}{7}$$

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

$$\frac{289}{784} - u^2 = -\frac{3}{7}$$

Simplify the expression by subtracting \frac{289}{784} on both sides

$$-u^2 = -\frac{3}{7}-\frac{289}{784} = -\frac{625}{784}$$

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

$$u^2 = \frac{625}{784} u = \pm\sqrt{\frac{625}{784}} = \pm \frac{25}{28}$$

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

$$r =-\frac{17}{28} - \frac{25}{28} = -1.500 s = -\frac{17}{28} + \frac{25}{28} = 0.286$$

Evaluate

$$\left(7x-2\right)\left(2x+3\right)$$