Solve 8x^2+13x-6

$$8 x ^ { 2 } + 13 x - 6$$

Factor

$$\left(8x-3\right)\left(x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8x^{2}+ax+bx-6. To find a and b, set up a system to be solved.

$$a+b=13 ab=8\left(-6\right)=-48$$

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 -48.

$$-1,48 -2,24 -3,16 -4,12 -6,8$$

Calculate the sum for each pair.

$$-1+48=47 -2+24=22 -3+16=13 -4+12=8 -6+8=2$$

The solution is the pair that gives sum 13.

$$a=-3 b=16$$

Rewrite 8x^{2}+13x-6 as \left(8x^{2}-3x\right)+\left(16x-6\right).

$$\left(8x^{2}-3x\right)+\left(16x-6\right)$$

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

$$x\left(8x-3\right)+2\left(8x-3\right)$$

Factor out common term 8x-3 by using distributive property.

$$\left(8x-3\right)\left(x+2\right)$$

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.

$$8x^{2}+13x-6=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\times 8\left(-6\right)}}{2\times 8}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 8\left(-6\right)}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-13±\sqrt{169-32\left(-6\right)}}{2\times 8}$$

Multiply -32 times -6.

$$x=\frac{-13±\sqrt{169+192}}{2\times 8}$$

Add 169 to 192.

$$x=\frac{-13±\sqrt{361}}{2\times 8}$$

Take the square root of 361.

$$x=\frac{-13±19}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-13±19}{16}$$

Now solve the equation x=\frac{-13±19}{16} when ± is plus. Add -13 to 19.

$$x=\frac{6}{16}$$

Reduce the fraction \frac{6}{16} to lowest terms by extracting and canceling out 2.

$$x=\frac{3}{8}$$

Now solve the equation x=\frac{-13±19}{16} when ± is minus. Subtract 19 from -13.

$$x=-\frac{32}{16}$$

Divide -32 by 16.

$$x=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{3}{8} for x_{1} and -2 for x_{2}.

$$8x^{2}+13x-6=8\left(x-\frac{3}{8}\right)\left(x-\left(-2\right)\right)$$

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

$$8x^{2}+13x-6=8\left(x-\frac{3}{8}\right)\left(x+2\right)$$

Subtract \frac{3}{8} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}+13x-6=8\times \frac{8x-3}{8}\left(x+2\right)$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8x^{2}+13x-6=\left(8x-3\right)\left(x+2\right)$$

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 8

$$x ^ 2 +\frac{13}{8}x -\frac{3}{4} = 0$$

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

$$r + s = -\frac{13}{8} rs = -\frac{3}{4}$$

Two numbers r and s sum up to -\frac{13}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{8} = -\frac{13}{16}. 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>

$$r = -\frac{13}{16} - u s = -\frac{13}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{3}{4}

$$(-\frac{13}{16} - u) (-\frac{13}{16} + u) = -\frac{3}{4}$$

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

$$\frac{169}{256} - u^2 = -\frac{3}{4}$$

Simplify the expression by subtracting \frac{169}{256} on both sides

$$-u^2 = -\frac{3}{4}-\frac{169}{256} = -\frac{361}{256}$$

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

$$u^2 = \frac{361}{256} u = \pm\sqrt{\frac{361}{256}} = \pm \frac{19}{16}$$

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

$$r =-\frac{13}{16} - \frac{19}{16} = -2 s = -\frac{13}{16} + \frac{19}{16} = 0.375$$

Evaluate

$$\left(8x-3\right)\left(x+2\right)$$