First we need to factor trinomial $ \color{blue}{ -x^2+5x-6 } $ and than we use factored form to solve an equation $ \color{blue}{ -x^2+5x-6 = 0} $.

Step 1: We can simplify equation by multiplying both sides by -1. After multiplying we have the following equation:

$$ \begin{aligned} -x^2+5x-6 &= 0 \,\,\, / \color{orangered}{\cdot \, -1 } \\[0.9 em ] x^2-5x+6 &=0 \end{aligned} $$

Step 1: Identify constants $ \color{blue}{ b }$ and $\color{red}{ c }$. ( $ \color{blue}{ b }$ is a number in front of the $ x $ term and $ \color{red}{ c } $ is a constant). In our case:

$$ \color{blue}{ b = -5 } ~ \text{ and } ~ \color{red}{ c = 6 }$$

Now we must discover two numbers that sum up to $ \color{blue}{ -5 } $ and multiply to $ \color{red}{ 6 } $.

Step 2: Find out pairs of numbers with a product of $\color{red}{ c = 6 }$.

PRODUCT = 6
1     6	-1     -6
2     3	-2     -3

Step 3: Find out which pair sums up to $\color{blue}{ b = -5 }$

PRODUCT = 6 and SUM = -5
1     6	-1     -6
2     3	-2     -3

Step 4: Put -2 and -3 into placeholders to get factored form.

$$ \begin{aligned} x^{2}-5x+6 & = (x + \color{orangered}{\square} )(x + \color{orangered}{\square}) \\ x^{2}-5x+6 & = (x -2)(x -3) \end{aligned} $$

Step 5: Set each factor to zero and solve equations.

$$ \begin{array}{ccc} \begin{aligned} x-2 &= 0 \\ x &= 2 \end{aligned} & ~ & \begin{aligned} x-3 &= 0 \\ x &= 3 \end{aligned} \end{array} $$

Quadratic Equation Solver