Step-by-Step Math Solver with Hints & Practice

A. Goal:
Find all real solutions ( x ) to the quadratic equation ( x^2 - 5x + 6 = 0 ).

B. Step-by-step solution:

Method 1: Factoring

1. Write the quadratic equation:
   [ x^2 - 5x + 6 = 0 ]
   Reason: Starting point, given equation.

2. Find two numbers that multiply to ( c = 6 ) and add to ( b = -5 ):
   Numbers are (-2) and (-3) because ((-2) \times (-3) = 6) and ((-2) + (-3) = -5).
   Reason: Factoring trinomial into two binomials.

3. Factor the quadratic:
   [ (x - 2)(x - 3) = 0 ]
   Reason: Express quadratic as product of two factors.

4. Apply the zero product property:
   Set each factor to zero:
   [ x - 2 = 0 \quad \Rightarrow \quad x = 2 ]
   [ x - 3 = 0 \quad \Rightarrow \quad x = 3 ]
   Reason: If product is zero, one of the factors must be zero.

Method 2: Quadratic Formula

1. Write given coefficients:
   [ a = 1, \quad b = -5, \quad c = 6 ]
   Reason: Identify coefficients for formula.

2. Write quadratic formula:
   [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]
   Reason: Formula solves any quadratic equation.

3. Calculate discriminant:
   [ \Delta = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1 ]
   Reason: Discriminant determines nature of roots.

4. Apply formula:
   [ x = \frac{-(-5) \pm \sqrt{1}}{2 \times 1} = \frac{5 \pm 1}{2} ]
   Reason: Substitute values into formula.

5. Calculate solutions:
   [ x1 = \frac{5 + 1}{2} = 3, \quad x2 = \frac{5 - 1}{2} = 2 ]
   Reason: Find exact roots of quadratic.

C. Verification:

Check both values in original equation ( x^2 - 5x + 6 = 0 ):

• For ( x = 2 ): [ 2^2 - 5(2) + 6 = 4 - 10 + 6 = 0 ]
• For ( x = 3 ): [ 3^2 - 5(3) + 6 = 9 - 15 + 6 = 0 ]

Both satisfy the equation.

D. Final answer:

[ \boxed{x = 2 \text{ or } x = 3} ]

E. Additional methods:

• Completing the square (not required here but a useful method).
• Graphical method (finding roots as ( x )-intercepts of ( y = x^2 - 5x + 6 )).

F. Key learning points:

1. Always check signs carefully when factoring to avoid mistakes.
2. Verify solutions by substituting back into the original equation.

G. Practice problems:

1. Solve ( x^2 - 7x + 10 = 0 )
   Short answer: ( x = 2 ) or ( x = 5 )

2. Solve ( x^2 + 2x - 8 = 0 )
   Short answer: ( x = 2 ) or ( x = -4 )

3. Solve ( 2x^2 - 3x - 2 = 0 )
   Short answer: ( x = 2 ) or ( x = -\frac{1}{2} )