Solve 2a^2-29a+54

$$2 a ^ { 2 } - 29 a + 54$$

Factor

$$2\left(a-\frac{29-\sqrt{409}}{4}\right)\left(a-\frac{\sqrt{409}+29}{4}\right)$$

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.

$$2a^{2}-29a+54=0$$

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.

$$a=\frac{-\left(-29\right)±\sqrt{\left(-29\right)^{2}-4\times 2\times 54}}{2\times 2}$$

Square -29.

$$a=\frac{-\left(-29\right)±\sqrt{841-4\times 2\times 54}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-\left(-29\right)±\sqrt{841-8\times 54}}{2\times 2}$$

Multiply -8 times 54.

$$a=\frac{-\left(-29\right)±\sqrt{841-432}}{2\times 2}$$

Add 841 to -432.

$$a=\frac{-\left(-29\right)±\sqrt{409}}{2\times 2}$$

The opposite of -29 is 29.

$$a=\frac{29±\sqrt{409}}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{29±\sqrt{409}}{4}$$

Now solve the equation a=\frac{29±\sqrt{409}}{4} when ± is plus. Add 29 to \sqrt{409}.

$$a=\frac{\sqrt{409}+29}{4}$$

Now solve the equation a=\frac{29±\sqrt{409}}{4} when ± is minus. Subtract \sqrt{409} from 29.

$$a=\frac{29-\sqrt{409}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{29+\sqrt{409}}{4} for x_{1} and \frac{29-\sqrt{409}}{4} for x_{2}.

$$2a^{2}-29a+54=2\left(a-\frac{\sqrt{409}+29}{4}\right)\left(a-\frac{29-\sqrt{409}}{4}\right)$$

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

$$x ^ 2 -\frac{29}{2}x +27 = 0$$

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

$$r + s = \frac{29}{2} rs = 27$$

Two numbers r and s sum up to \frac{29}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{29}{2} = \frac{29}{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>

$$r = \frac{29}{4} - u s = \frac{29}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 27

$$(\frac{29}{4} - u) (\frac{29}{4} + u) = 27$$

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

$$\frac{841}{16} - u^2 = 27$$

Simplify the expression by subtracting \frac{841}{16} on both sides

$$-u^2 = 27-\frac{841}{16} = -\frac{409}{16}$$

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

$$u^2 = \frac{409}{16} u = \pm\sqrt{\frac{409}{16}} = \pm \frac{\sqrt{409}}{4}$$

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

$$r =\frac{29}{4} - \frac{\sqrt{409}}{4} = 2.194 s = \frac{29}{4} + \frac{\sqrt{409}}{4} = 12.306$$

Evaluate

$$2a^{2}-29a+54$$