Quadratic equations

Unit 2 Quadratic equations

Key unit competence: Solve problems by formulating and manipulating quadratic equations.

Key ideas

• A quadratic equation has the form $a{x}^2+bx+c=0$ with $a\ne 0$. It has at most two solutions (roots).
• Factorisation: write $a{x}^2+bx+c$ as a product, then use: if $pq=0$ then $p=0$ or $q=0$.
• Perfect squares: $(x+k{)}^2=m$ gives $x+k=\pm \sqrt{m}$.
• Completing the square: $x^2+bx={\left(x+\frac{b}{2}\right)}^2-{\left(\frac{b}{2}\right)}^2$.
• Quadratic formula: $x=\frac{-b\pm \sqrt{\Delta }}{2a}$ where the discriminant $\Delta =b^2-4ac$. $\Delta >0$: two distinct real roots; $\Delta =0$: one repeated root; $\Delta <0$: no real roots.
• Synthetic division: if one root $r$ is known, divide by $x-r$ to find the other factor.
• Sum and product of roots: $x_1+x_2=-\frac{b}{a}$, $x_1{x}_2=\frac{c}{a}$.

Formulas to know

Common misconceptions

• "$x^2=9$ gives $x=3$ only." No: $x=3$ or $x=-3$.
• "From $x(x-5)=0$, divide by $x$ to get $x=5$." No: dividing by $x$ loses the root $x=0$.
• "$(x-2)(x+3)=6$ gives $x-2=6$ or $x+3=6$." No: the product must equal zero to split into factors. Expand first.

Worked examples

Practice questions

Easier and harder questions are mixed, just like in a real exam.

1. Solve $x^2+2x-15=0$ by factorisation.

   Show answer

   $(x+5)(x-3)=0$, so $x=-5$ or $x=3$.

   Common mistake: Choosing factors with the wrong signs.

2. A garden 10 m by 6 m has a path of uniform width $x$ around it. The total area is 96 m². Find $x$.

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   $(10+2x)(6+2x)=96$: $4x^2+32x+60=96$, $x^2+8x-9=0$, $(x+9)(x-1)=0$, so $x=1$ m.

   Common mistake: Adding $x$ only once to each dimension.

3. Solve $x^2+6x+2=0$ by completing the square (exact answer).

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   $(x+3{)}^2=7$, $x=-3\pm \sqrt{7}$.

   Common mistake: Adding 9 to only one side.

4. Find two numbers whose sum is 10 and product is 21.

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   They are roots of $x^2-10x+21=0$: 3 and 7.

   Common mistake: Writing the equation as $x^2+10x+21=0$.

5. Solve $3x^2-x-2=0$ using the formula.

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   $\Delta =1+24=25$; $x=\frac{1\pm 5}{6}$: $x=1$ or $x=-\frac{2}{3}$.

   Common mistake: Writing $-b=-1$ instead of $+1$.

6. Solve $(2x-1{)}^2=9$.

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   $2x-1=\pm 3$: $x=2$ or $x=-1$.

   Common mistake: Taking only $+3$.

7. Determine the nature of the roots of $2x^2-3x+4=0$.

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   $\Delta =9-32=-23<0$: no real roots.

   Common mistake: Saying "no solution" without computing $\Delta$.

8. The hypotenuse of a right triangle is 13 cm; one leg is 7 cm longer than the other. Find the legs.

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   $x^2+(x+7{)}^2=169$, $2x^2+14x-120=0$, $x^2+7x-60=0$, $(x+12)(x-5)=0$: legs 5 cm and 12 cm.

   Common mistake: Keeping the negative root.

Matching and drag-and-drop

1. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Solve $x^2-x-12=0$.	a. $x=4$ or $x=-3$
   2. $\alpha$ and $\beta$ are the roots of $x^2-5x+3=0$. Find ${\alpha }^2+{\beta }^2$.	b. $x=0$ or $x=5$
   3. Solve $3x^2-5x-2=0$.	c. $x=2$ or $x=-\frac{1}{3}$
   4. Solve $x^2-5x=0$.	d. 19

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   1 → a ($x=4$ or $x=-3$) · 2 → d (19) · 3 → c ($x=2$ or $x=-\frac{1}{3}$) · 4 → b ($x=0$ or $x=5$)

   • $(x-4)(x+3)=0$.
   • Step 1. $\alpha +\beta =5$ and $\alpha \beta =3$.
     Step 2. ${\alpha }^2+{\beta }^2=(\alpha +\beta {)}^2-2\alpha \beta =25-6=19$.
   • Step 1. $ac=-6$: numbers $-6$ and 1. $3x^2-6x+x-2=3x(x-2)+(x-2)$.
     Step 2. $(3x+1)(x-2)=0$: $x=2$ or $x=-\frac{1}{3}$.
   • $x(x-5)=0$.

2. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   $-6$  ·  one repeated root $x=3$  ·  $-3$  ·  $-2$
   1. The product of the roots of $2x^2+5x-12=0$ is .
   2. $x^2-6x+9=0$ has .
   3. One root of $x^2-x-6=0$ is 3. The other is .
   Show answer

   1: $-6$ · 2: one repeated root $x=3$ · 3: $-2$

   • $\frac{c}{a}=\frac{-12}{2}$.
   • Step 1. $(x-3{)}^2=0$.
     Step 2. $\Delta =36-36=0$.
   • Step 1. Product of roots $=-6$.
     Step 2. So $3\times x_2=-6$.

3. Matching

   Match each item in column A with the correct answer in column B. Write the letter next to each number.

   Column A	Column B
   1. Solve $x^2+7x+10=0$.	a. 9
   2. Find the discriminant of $x^2+4x+1=0$.	b. 12
   3. Complete the square: $x^2+6x=(x+3{)}^2-?$	c. $x=2\pm \sqrt{5}$
   4. Solve $x^2-4x-1=0$.	d. $x=-2$ or $x=-5$

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   1 → d ($x=-2$ or $x=-5$) · 2 → b (12) · 3 → a (9) · 4 → c ($x=2\pm \sqrt{5}$)

   • $(x+2)(x+5)=0$.
   • $16-4=12$.
   • ${\left(\frac{6}{2}\right)}^2=9$.
   • Step 1. $\Delta =16+4=20$.
     Step 2. $x=\frac{4\pm \sqrt{20}}{2}=\frac{4\pm 2\sqrt{5}}{2}=2\pm \sqrt{5}$.

4. Drag and drop

   Choose the correct word or phrase from the box for each gap. One word in the box is not needed.

   7  ·  $2(x-2{)}^2+3$  ·  $t=4$ s  ·  $2(x-2{)}^2-5$
   1. The sum of the roots of $x^2-7x+10=0$ is .
   2. A ball's height is $h=20t-5t^2$ metres after $t$ seconds. When is it back on the ground? →
   3. Write $2x^2-8x+3$ in the form $a(x-p{)}^2+q$. →
   Show answer

   1: 7 · 2: $t=4$ s · 3: $2(x-2{)}^2-5$

   • $-\frac{b}{a}=7$ (roots 2 and 5).
   • $5t(4-t)=0$ gives $t=0$ (start) or $t=4$.
   • Step 1. $2(x^2-4x)+3=2\left[(x-2{)}^2-4\right]+3$.
     Step 2. $=2(x-2{)}^2-8+3=2(x-2{)}^2-5$.

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