Solve 3x^2-14x-5

$$3 x ^ { 2 } - 14 x - 5$$

Factor

$$\left(x-5\right)\left(3x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3x^{2}+ax+bx-5. To find a and b, set up a system to be solved.

$$a+b=-14 ab=3\left(-5\right)=-15$$

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 -15.

$$1,-15 3,-5$$

Calculate the sum for each pair.

$$1-15=-14 3-5=-2$$

The solution is the pair that gives sum -14.

$$a=-15 b=1$$

Rewrite 3x^{2}-14x-5 as \left(3x^{2}-15x\right)+\left(x-5\right).

$$\left(3x^{2}-15x\right)+\left(x-5\right)$$

Factor out 3x in 3x^{2}-15x.

$$3x\left(x-5\right)+x-5$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(3x+1\right)$$

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.

$$3x^{2}-14x-5=0$$

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.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 3\left(-5\right)}}{2\times 3}$$

Square -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 3\left(-5\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-14\right)±\sqrt{196-12\left(-5\right)}}{2\times 3}$$

Multiply -12 times -5.

$$x=\frac{-\left(-14\right)±\sqrt{196+60}}{2\times 3}$$

Add 196 to 60.

$$x=\frac{-\left(-14\right)±\sqrt{256}}{2\times 3}$$

Take the square root of 256.

$$x=\frac{-\left(-14\right)±16}{2\times 3}$$

The opposite of -14 is 14.

$$x=\frac{14±16}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{14±16}{6}$$

Now solve the equation x=\frac{14±16}{6} when ± is plus. Add 14 to 16.

$$x=\frac{30}{6}$$

Divide 30 by 6.

$$x=5$$

Now solve the equation x=\frac{14±16}{6} when ± is minus. Subtract 16 from 14.

$$x=-\frac{2}{6}$$

Reduce the fraction \frac{-2}{6} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 5 for x_{1} and -\frac{1}{3} for x_{2}.

$$3x^{2}-14x-5=3\left(x-5\right)\left(x-\left(-\frac{1}{3}\right)\right)$$

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

$$3x^{2}-14x-5=3\left(x-5\right)\left(x+\frac{1}{3}\right)$$

Add \frac{1}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$3x^{2}-14x-5=3\left(x-5\right)\times \frac{3x+1}{3}$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3x^{2}-14x-5=\left(x-5\right)\left(3x+1\right)$$

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 3

$$x ^ 2 -\frac{14}{3}x -\frac{5}{3} = 0$$

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

$$r + s = \frac{14}{3} rs = -\frac{5}{3}$$

Two numbers r and s sum up to \frac{14}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{14}{3} = \frac{7}{3}. 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>

$$r = \frac{7}{3} - u s = \frac{7}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{5}{3}

$$(\frac{7}{3} - u) (\frac{7}{3} + u) = -\frac{5}{3}$$

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

$$\frac{49}{9} - u^2 = -\frac{5}{3}$$

Simplify the expression by subtracting \frac{49}{9} on both sides

$$-u^2 = -\frac{5}{3}-\frac{49}{9} = -\frac{64}{9}$$

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

$$u^2 = \frac{64}{9} u = \pm\sqrt{\frac{64}{9}} = \pm \frac{8}{3}$$

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

$$r =\frac{7}{3} - \frac{8}{3} = -0.333 s = \frac{7}{3} + \frac{8}{3} = 5$$

Evaluate

$$\left(x-5\right)\left(3x+1\right)$$