Solve 8x^2+26x+20

$$8 x ^ { 2 } + 26 x + 20$$

Factor

$$2\left(x+2\right)\left(4x+5\right)$$

Solution Steps

Factor out 2.

$$2\left(4x^{2}+13x+10\right)$$

Consider 4x^{2}+13x+10. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=13 ab=4\times 10=40$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 40.

$$1,40 2,20 4,10 5,8$$

Calculate the sum for each pair.

$$1+40=41 2+20=22 4+10=14 5+8=13$$

The solution is the pair that gives sum 13.

$$a=5 b=8$$

Rewrite 4x^{2}+13x+10 as \left(4x^{2}+5x\right)+\left(8x+10\right).

$$\left(4x^{2}+5x\right)+\left(8x+10\right)$$

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

$$x\left(4x+5\right)+2\left(4x+5\right)$$

Factor out common term 4x+5 by using distributive property.

$$\left(4x+5\right)\left(x+2\right)$$

Rewrite the complete factored expression.

$$2\left(4x+5\right)\left(x+2\right)$$

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.

$$8x^{2}+26x+20=0$$

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.

$$x=\frac{-26±\sqrt{26^{2}-4\times 8\times 20}}{2\times 8}$$

Square 26.

$$x=\frac{-26±\sqrt{676-4\times 8\times 20}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-26±\sqrt{676-32\times 20}}{2\times 8}$$

Multiply -32 times 20.

$$x=\frac{-26±\sqrt{676-640}}{2\times 8}$$

Add 676 to -640.

$$x=\frac{-26±\sqrt{36}}{2\times 8}$$

Take the square root of 36.

$$x=\frac{-26±6}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-26±6}{16}$$

Now solve the equation x=\frac{-26±6}{16} when ± is plus. Add -26 to 6.

$$x=-\frac{20}{16}$$

Reduce the fraction \frac{-20}{16} to lowest terms by extracting and canceling out 4.

$$x=-\frac{5}{4}$$

Now solve the equation x=\frac{-26±6}{16} when ± is minus. Subtract 6 from -26.

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

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{5}{4} for x_{1} and -2 for x_{2}.

$$8x^{2}+26x+20=8\left(x-\left(-\frac{5}{4}\right)\right)\left(x-\left(-2\right)\right)$$

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

$$8x^{2}+26x+20=8\left(x+\frac{5}{4}\right)\left(x+2\right)$$

Add \frac{5}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}+26x+20=8\times \frac{4x+5}{4}\left(x+2\right)$$

Cancel out 4, the greatest common factor in 8 and 4.

$$8x^{2}+26x+20=2\left(4x+5\right)\left(x+2\right)$$

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

$$x ^ 2 +\frac{13}{4}x +\frac{5}{2} = 0$$

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

$$r + s = -\frac{13}{4} rs = \frac{5}{2}$$

Two numbers r and s sum up to -\frac{13}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{4} = -\frac{13}{8}. 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>

$$r = -\frac{13}{8} - u s = -\frac{13}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{2}

$$(-\frac{13}{8} - u) (-\frac{13}{8} + u) = \frac{5}{2}$$

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

$$\frac{169}{64} - u^2 = \frac{5}{2}$$

Simplify the expression by subtracting \frac{169}{64} on both sides

$$-u^2 = \frac{5}{2}-\frac{169}{64} = -\frac{9}{64}$$

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

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

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

$$r =-\frac{13}{8} - \frac{3}{8} = -2 s = -\frac{13}{8} + \frac{3}{8} = -1.250$$

Evaluate

$$2\left(x+2\right)\left(4x+5\right)$$