Solve 8x^2+65x+8

$$8 x ^ { 2 } + 65 x + 8$$

Factor

$$\left(x+8\right)\left(8x+1\right)$$

Solution Steps

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

$$a+b=65 ab=8\times 8=64$$

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 64.

$$1,64 2,32 4,16 8,8$$

Calculate the sum for each pair.

$$1+64=65 2+32=34 4+16=20 8+8=16$$

The solution is the pair that gives sum 65.

$$a=1 b=64$$

Rewrite 8x^{2}+65x+8 as \left(8x^{2}+x\right)+\left(64x+8\right).

$$\left(8x^{2}+x\right)+\left(64x+8\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(8x+1\right)+8\left(8x+1\right)$$

Factor out common term 8x+1 by using distributive property.

$$\left(8x+1\right)\left(x+8\right)$$

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.

$$8x^{2}+65x+8=0$$

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.

$$x=\frac{-65±\sqrt{65^{2}-4\times 8\times 8}}{2\times 8}$$

Square 65.

$$x=\frac{-65±\sqrt{4225-4\times 8\times 8}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-65±\sqrt{4225-32\times 8}}{2\times 8}$$

Multiply -32 times 8.

$$x=\frac{-65±\sqrt{4225-256}}{2\times 8}$$

Add 4225 to -256.

$$x=\frac{-65±\sqrt{3969}}{2\times 8}$$

Take the square root of 3969.

$$x=\frac{-65±63}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{-65±63}{16}$$

Now solve the equation x=\frac{-65±63}{16} when ± is plus. Add -65 to 63.

$$x=-\frac{2}{16}$$

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

$$x=-\frac{1}{8}$$

Now solve the equation x=\frac{-65±63}{16} when ± is minus. Subtract 63 from -65.

$$x=-\frac{128}{16}$$

Divide -128 by 16.

$$x=-8$$

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

$$8x^{2}+65x+8=8\left(x-\left(-\frac{1}{8}\right)\right)\left(x-\left(-8\right)\right)$$

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

$$8x^{2}+65x+8=8\left(x+\frac{1}{8}\right)\left(x+8\right)$$

Add \frac{1}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8x^{2}+65x+8=8\times \frac{8x+1}{8}\left(x+8\right)$$

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

$$8x^{2}+65x+8=\left(8x+1\right)\left(x+8\right)$$

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

$$x ^ 2 +\frac{65}{8}x +1 = 0$$

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

$$r + s = -\frac{65}{8} rs = 1$$

Two numbers r and s sum up to -\frac{65}{8} exactly when the average of the two numbers is \frac{1}{2}*-\frac{65}{8} = -\frac{65}{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>

$$r = -\frac{65}{16} - u s = -\frac{65}{16} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 1

$$(-\frac{65}{16} - u) (-\frac{65}{16} + u) = 1$$

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

$$\frac{4225}{256} - u^2 = 1$$

Simplify the expression by subtracting \frac{4225}{256} on both sides

$$-u^2 = 1-\frac{4225}{256} = -\frac{3969}{256}$$

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

$$u^2 = \frac{3969}{256} u = \pm\sqrt{\frac{3969}{256}} = \pm \frac{63}{16}$$

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

$$r =-\frac{65}{16} - \frac{63}{16} = -8 s = -\frac{65}{16} + \frac{63}{16} = -0.125$$

Evaluate

$$\left(x+8\right)\left(8x+1\right)$$