Solve 4a^2-19a+17

$$4 a ^ { 2 } - 19 a + 17$$

Factor

$$4\left(a-\frac{19-\sqrt{89}}{8}\right)\left(a-\frac{\sqrt{89}+19}{8}\right)$$

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.

$$4a^{2}-19a+17=0$$

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.

$$a=\frac{-\left(-19\right)±\sqrt{\left(-19\right)^{2}-4\times 4\times 17}}{2\times 4}$$

Square -19.

$$a=\frac{-\left(-19\right)±\sqrt{361-4\times 4\times 17}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-\left(-19\right)±\sqrt{361-16\times 17}}{2\times 4}$$

Multiply -16 times 17.

$$a=\frac{-\left(-19\right)±\sqrt{361-272}}{2\times 4}$$

Add 361 to -272.

$$a=\frac{-\left(-19\right)±\sqrt{89}}{2\times 4}$$

The opposite of -19 is 19.

$$a=\frac{19±\sqrt{89}}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{19±\sqrt{89}}{8}$$

Now solve the equation a=\frac{19±\sqrt{89}}{8} when ± is plus. Add 19 to \sqrt{89}.

$$a=\frac{\sqrt{89}+19}{8}$$

Now solve the equation a=\frac{19±\sqrt{89}}{8} when ± is minus. Subtract \sqrt{89} from 19.

$$a=\frac{19-\sqrt{89}}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{19+\sqrt{89}}{8} for x_{1} and \frac{19-\sqrt{89}}{8} for x_{2}.

$$4a^{2}-19a+17=4\left(a-\frac{\sqrt{89}+19}{8}\right)\left(a-\frac{19-\sqrt{89}}{8}\right)$$

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 4

$$x ^ 2 -\frac{19}{4}x +\frac{17}{4} = 0$$

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

$$r + s = \frac{19}{4} rs = \frac{17}{4}$$

Two numbers r and s sum up to \frac{19}{4} exactly when the average of the two numbers is \frac{1}{2}*\frac{19}{4} = \frac{19}{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>

$$r = \frac{19}{8} - u s = \frac{19}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{17}{4}

$$(\frac{19}{8} - u) (\frac{19}{8} + u) = \frac{17}{4}$$

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

$$\frac{361}{64} - u^2 = \frac{17}{4}$$

Simplify the expression by subtracting \frac{361}{64} on both sides

$$-u^2 = \frac{17}{4}-\frac{361}{64} = -\frac{89}{64}$$

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

$$u^2 = \frac{89}{64} u = \pm\sqrt{\frac{89}{64}} = \pm \frac{\sqrt{89}}{8}$$

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

$$r =\frac{19}{8} - \frac{\sqrt{89}}{8} = 1.196 s = \frac{19}{8} + \frac{\sqrt{89}}{8} = 3.554$$

Evaluate

$$4a^{2}-19a+17$$