Solve f(x)=-3x^2+19x-2

$$f ( x ) = - 3 x ^ { 2 } + 19 x - 2$$

Factor

$$-3\left(x-\frac{19-\sqrt{337}}{6}\right)\left(x-\frac{\sqrt{337}+19}{6}\right)$$

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.

$$-3x^{2}+19x-2=0$$

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.

$$x=\frac{-19±\sqrt{19^{2}-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

Square 19.

$$x=\frac{-19±\sqrt{361-4\left(-3\right)\left(-2\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-19±\sqrt{361+12\left(-2\right)}}{2\left(-3\right)}$$

Multiply 12 times -2.

$$x=\frac{-19±\sqrt{361-24}}{2\left(-3\right)}$$

Add 361 to -24.

$$x=\frac{-19±\sqrt{337}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-19±\sqrt{337}}{-6}$$

Now solve the equation x=\frac{-19±\sqrt{337}}{-6} when ± is plus. Add -19 to \sqrt{337}.

$$x=\frac{\sqrt{337}-19}{-6}$$

Divide -19+\sqrt{337} by -6.

$$x=\frac{19-\sqrt{337}}{6}$$

Now solve the equation x=\frac{-19±\sqrt{337}}{-6} when ± is minus. Subtract \sqrt{337} from -19.

$$x=\frac{-\sqrt{337}-19}{-6}$$

Divide -19-\sqrt{337} by -6.

$$x=\frac{\sqrt{337}+19}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{19-\sqrt{337}}{6} for x_{1} and \frac{19+\sqrt{337}}{6} for x_{2}.

$$-3x^{2}+19x-2=-3\left(x-\frac{19-\sqrt{337}}{6}\right)\left(x-\frac{\sqrt{337}+19}{6}\right)$$

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.

$$x ^ 2 -\frac{19}{3}x +\frac{2}{3} = 0$$

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

$$r + s = \frac{19}{3} rs = \frac{2}{3}$$

Two numbers r and s sum up to \frac{19}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{19}{3} = \frac{19}{6}. 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>

$$r = \frac{19}{6} - u s = \frac{19}{6} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{2}{3}

$$(\frac{19}{6} - u) (\frac{19}{6} + u) = \frac{2}{3}$$

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

$$\frac{361}{36} - u^2 = \frac{2}{3}$$

Simplify the expression by subtracting \frac{361}{36} on both sides

$$-u^2 = \frac{2}{3}-\frac{361}{36} = -\frac{337}{36}$$

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

$$u^2 = \frac{337}{36} u = \pm\sqrt{\frac{337}{36}} = \pm \frac{\sqrt{337}}{6}$$

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

$$r =\frac{19}{6} - \frac{\sqrt{337}}{6} = 0.107 s = \frac{19}{6} + \frac{\sqrt{337}}{6} = 6.226$$

Evaluate

$$-3x^{2}+19x-2$$