Quadratic Equations

Solve $x^2+5x+6=0$.

• 1Find two numbers that multiply to $6$ and add to $5$. Those numbers are $2$ and $3$.
• 2Write as brackets: ($x+2$)($x+3$) = $0$
• 3Either bracket could be zero: $x+2=0$ or $x+3=0$
• 4Solve each: $x=-2$ or $x=-3$

Answer: $x=-2$ or $x=-3$

Solve $x^2-x-6=0$.

• 1Find two numbers that multiply to $-6$ and add to $-1$. Those numbers are $-3$ and $2$.
• 2Write as brackets: ($x-3$)($x+2$) = $0$
• 3Solve each bracket: $x-3=0$ or $x+2=0$
• 4$x=3$ or $x=-2$

Answer: $x=3$ or $x=-2$

Solve $x^2-6x+9=0$.

• 1Find two numbers that multiply to $9$ and add to $-6$. Those numbers are $-3$ and $-3$.
• 2Write as brackets: ($x-3$)($x-3$) = $0$
• 3Both brackets give the same equation: $x=3$

Answer: $x=3$ (a repeated solution)

Solve $2x^2+7x+3=0$.

• 1Multiply a and c: $2\times 3=6$. Find two numbers that multiply to $6$ and add to $7$: those are $1$ and $6$.
• 2Split the middle term: $2x^2+x+6x+3=0$
• 3Factor in pairs: $x$($2x+1$) + $3$($2x+1$) = $0$
• 4Write as brackets: ($2x+1$)($x+3$) = $0$
• 5Solve each: $2x+1=0$, so $x=-0.5$; or $x+3=0$, so $x=-3$

Answer: $x=-0.5$ or $x=-3$