Solve 0x^2-5x+6=0

$$0 x ^ { 2 } - 5 x + 6 = 0$$

Solve for x

$$x = \frac{6}{5} = 1\frac{1}{5} = 1.2$$

Steps for Solving Linear Equation

Anything times zero gives zero.

$$0-5x+6=0$$

Add 0 and 6 to get 6.

$$6-5x=0$$

Subtract 6 from both sides. Anything subtracted from zero gives its negation.

$$-5x=-6$$

Divide both sides by -5.

$$x=\frac{-6}{-5}$$

Fraction \frac{-6}{-5} can be simplified to \frac{6}{5} by removing the negative sign from both the numerator and the denominator.

$$x=\frac{6}{5}$$

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.

$$x ^ 2 -5x +6 = 0$$

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

$$r + s = 5 rs = 6$$

Two numbers r and s sum up to 5 exactly when the average of the two numbers is \frac{1}{2}*5 = \frac{5}{2}. 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>

$$r = \frac{5}{2} - u s = \frac{5}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 6

$$(\frac{5}{2} - u) (\frac{5}{2} + u) = 6$$

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

$$\frac{25}{4} - u^2 = 6$$

Simplify the expression by subtracting \frac{25}{4} on both sides

$$-u^2 = 6-\frac{25}{4} = -\frac{1}{4}$$

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

$$u^2 = \frac{1}{4} u = \pm\sqrt{\frac{1}{4}} = \pm \frac{1}{2}$$

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

$$r =\frac{5}{2} - \frac{1}{2} = 2 s = \frac{5}{2} + \frac{1}{2} = 3$$