Solve the equation x² − 5x + 6 = 0
Answer: \(x=2\) or \(x=3\).
Key steps
Factor the quadratic:
\[ x^2-5x+6=0 \]
Find two numbers whose:
- product is \(6\);
- sum is \(-5\).
These numbers are \(-2\) and \(-3\), so:
\[ (x-2)(x-3)=0 \]
By the zero-product rule, at least one factor must equal zero:
\[ x-2=0 \quad \text{or} \quad x-3=0 \]
Therefore:
\[ \boxed{x=2 \text{ or } x=3} \]
Check
\[ 2^2-5\cdot2+6=4-10+6=0 \]
\[ 3^2-5\cdot3+6=9-15+6=0 \]
Both solutions are correct.
Обсуждение
2 сообщенияHow do I solve it using the quadratic formula?
Use the quadratic formula:
\[ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \]
For
\[ x^2-5x+6=0, \]
the coefficients are:
\[ a=1,\qquad b=-5,\qquad c=6. \]
Substitute them into the formula:
\[ x=\frac{-(-5)\pm\sqrt{(-5)^2-4(1)(6)}}{2(1)} \]
Simplify:
\[ x=\frac{5\pm\sqrt{25-24}}{2} =\frac{5\pm1}{2}. \]
Now calculate both solutions:
\[ x_1=\frac{5+1}{2}=3, \]
\[ x_2=\frac{5-1}{2}=2. \]
Therefore,
\[ \boxed{x=2 \text{ or } x=3}. \]
