Faded examples · Solving a quadratic with the formula (surd form)
Each example shows a little less than the one before – complete the faded steps yourself, using the same three steps every time: write $a,b,c$; substitute into the formula; simplify under the root. Also answer the check question. No calculator needed (leave answers in surd form).
①Example 1fully worked: read it through
Solve $x^2+3x+1=0$ in surd form.
1Write $a$, $b$, $c$
$a=1,\ b=3,\ c=1$
2Substitute into $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{-3\pm\sqrt{9-4}}{2}$
3Simplify under the root
$=\dfrac{-3\pm\sqrt{5}}{2}$
Check · Why is the first term $-3$ and not $+3$?A the formula starts with $-b$, and $b=3$B because $3$ is oddC a typing error – it should be $+3$D because $c=1$
②Example 2you finish the last 1 step
Solve $x^2+5x+3=0$ in surd form.
1Write $a$, $b$, $c$
$a=1,\ b=5,\ c=3$
2Substitute into $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{-5\pm\sqrt{25-12}}{2}$
3Simplify under the root
$=\dfrac{-5\pm\sqrt{13}}{2}$
Check · What is $b^2-4ac$ here?A $25+12=37$B $25-12=13$C $5-12=-7$D $10-12=-2$
③Example 3you finish the last 2 steps
Solve $x^2-x-1=0$ in surd form.
1Write $a$, $b$, $c$
$a=1,\ b=-1,\ c=-1$
2Substitute into $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{1\pm\sqrt{1+4}}{2}$
3Simplify under the root
$=\dfrac{1\pm\sqrt{5}}{2}$
Check · Why does $-4ac$ become $+4$?A $-4\times1\times(-1)=+4$ (two negatives)B a mistake – it should be $-4$C because $b=-1$D because the root is positive
④Example 4your turn: every step
Solve $x^2+3x-1=0$ in surd form.
1Write $a$, $b$, $c$
$a=1,\ b=3,\ c=-1$
2Substitute into $\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$
$x=\dfrac{-3\pm\sqrt{9+4}}{2}$
3Simplify under the root
$=\dfrac{-3\pm\sqrt{13}}{2}$
Check · The two roots come from the $\pm$. What does that mean here?A only take the $+$ signB there are two roots: $\dfrac{-3+\sqrt{13}}{2}$ and $\dfrac{-3-\sqrt{13}}{2}$C the answer is a single numberD the $\pm$ can be ignored
Answers · The Quadratic Formula
Faded examples · Solving a quadratic with the formula (surd form)
① Example 1 $a=1,\ b=3,\ c=1$→$x=\dfrac{-3\pm\sqrt{9-4}}{2}$→$=\dfrac{-3\pm\sqrt{5}}{2}$ $\dfrac{-3\pm\sqrt{5}}{2}$Check: A: the formula starts with $-b$, and $b=3$
② Example 2 $a=1,\ b=5,\ c=3$→$x=\dfrac{-5\pm\sqrt{25-12}}{2}$→$=\dfrac{-5\pm\sqrt{13}}{2}$ $\dfrac{-5\pm\sqrt{13}}{2}$Check: B: $25-12=13$
③ Example 3 $a=1,\ b=-1,\ c=-1$→$x=\dfrac{1\pm\sqrt{1+4}}{2}$→$=\dfrac{1\pm\sqrt{5}}{2}$ $\dfrac{1\pm\sqrt{5}}{2}$Check: A: $-4\times1\times(-1)=+4$ (two negatives)
④ Example 4 $a=1,\ b=3,\ c=-1$→$x=\dfrac{-3\pm\sqrt{9+4}}{2}$→$=\dfrac{-3\pm\sqrt{13}}{2}$ $\dfrac{-3\pm\sqrt{13}}{2}$Check: B: there are two roots: $\dfrac{-3+\sqrt{13}}{2}$ and $\dfrac{-3-\sqrt{13}}{2}$
mathedup.co.uk · sheet B3IA