Solve the equation x² − 5x + 6 = 0

Answer: \(x_{1}=2,\; x_{2}=3\).

The photo is too out of focus to read, so I solved the equation from the text you typed.

Key idea: a product equals zero

Write the quadratic as a product of two brackets. You need two numbers that multiply to \(+6\) and add to \(-5\) (because the middle term is \(-5x\)):

\[ x^{2}-5x+6=(x-2)(x-3) \]

Check the expansion: \(x\cdot x=x^{2}\), the outer and inner terms give \(-3x-2x=-5x\), and \((-2)(-3)=6\). ✓

Solve

\[ (x-2)(x-3)=0 \]

A product is zero only when one of the factors is zero:

\[ x-2=0 \Rightarrow x=2, \qquad x-3=0 \Rightarrow x=3 \]

Backup method: discriminant

\[ D=(-5)^{2}-4\cdot 1\cdot 6=25-24=1 \] \[ x=\frac{5\pm\sqrt{1}}{2}=\frac{5\pm 1}{2} \] so \(x=3\) or \(x=2\) — same result.

Check

• \(x=2:\; 4-10+6=0\) ✓
• \(x=3:\; 9-15+6=0\) ✓
• Vieta: \(x_{1}+x_{2}=5\) and \(x_{1}x_{2}=6\) ✓

How to use this next time

For \(x^{2}+bx+c=0\) with leading coefficient 1, look for two numbers with sum \(=-b\) and product \(=c\). If no such integers exist, switch to the discriminant formula \(x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).