Solve {l}{2x^2-13x}{-7}

$$\left. \begin{array} { l } { 2 x ^ { 2 } - 13 x } \\ { - 7 } \end{array} \right.$$

Factor

$$\left(x-7\right)\left(2x+1\right)$$

Solution Steps

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

$$a+b=-13 ab=2\left(-7\right)=-14$$

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 -14.

$$1,-14 2,-7$$

Calculate the sum for each pair.

$$1-14=-13 2-7=-5$$

The solution is the pair that gives sum -13.

$$a=-14 b=1$$

Rewrite 2x^{2}-13x-7 as \left(2x^{2}-14x\right)+\left(x-7\right).

$$\left(2x^{2}-14x\right)+\left(x-7\right)$$

Factor out 2x in 2x^{2}-14x.

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

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

$$\left(x-7\right)\left(2x+1\right)$$

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.

$$2x^{2}-13x-7=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 2\left(-7\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-13\right)±\sqrt{169-8\left(-7\right)}}{2\times 2}$$

Multiply -8 times -7.

$$x=\frac{-\left(-13\right)±\sqrt{169+56}}{2\times 2}$$

Add 169 to 56.

$$x=\frac{-\left(-13\right)±\sqrt{225}}{2\times 2}$$

Take the square root of 225.

$$x=\frac{-\left(-13\right)±15}{2\times 2}$$

The opposite of -13 is 13.

$$x=\frac{13±15}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{13±15}{4}$$

Now solve the equation x=\frac{13±15}{4} when ± is plus. Add 13 to 15.

$$x=\frac{28}{4}$$

Divide 28 by 4.

$$x=7$$

Now solve the equation x=\frac{13±15}{4} when ± is minus. Subtract 15 from 13.

$$x=-\frac{2}{4}$$

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

$$x=-\frac{1}{2}$$

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

$$2x^{2}-13x-7=2\left(x-7\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

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

$$2x^{2}-13x-7=2\left(x-7\right)\left(x+\frac{1}{2}\right)$$

Add \frac{1}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2x^{2}-13x-7=2\left(x-7\right)\times \frac{2x+1}{2}$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2x^{2}-13x-7=\left(x-7\right)\left(2x+1\right)$$

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 2

$$x ^ 2 -\frac{13}{2}x -\frac{7}{2} = 0$$

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

$$r + s = \frac{13}{2} rs = -\frac{7}{2}$$

Two numbers r and s sum up to \frac{13}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{13}{2} = \frac{13}{4}. 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>

$$r = \frac{13}{4} - u s = \frac{13}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{2}

$$(\frac{13}{4} - u) (\frac{13}{4} + u) = -\frac{7}{2}$$

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

$$\frac{169}{16} - u^2 = -\frac{7}{2}$$

Simplify the expression by subtracting \frac{169}{16} on both sides

$$-u^2 = -\frac{7}{2}-\frac{169}{16} = -\frac{225}{16}$$

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

$$u^2 = \frac{225}{16} u = \pm\sqrt{\frac{225}{16}} = \pm \frac{15}{4}$$

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

$$r =\frac{13}{4} - \frac{15}{4} = -0.500 s = \frac{13}{4} + \frac{15}{4} = 7$$

Evaluate

$$\left(x-7\right)\left(2x+1\right)$$