Quadratic equations

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1. Solve quadratic equations by taking square roots

Warm-up

Three quick questions from earlier lessons.

Discover

A square rug has an area of 49 square feet. Which numbers xx make x2=49x^2 = 49 true?

  1. 77 and −7-7
  2. Only 77

Key idea

Undo a square with ±.

If x2=kx^2 = k with k>0k > 0, then x=kx = \sqrt{k} or x=−kx = -\sqrt{k}. If (x−h)2=k(x - h)^2 = k, then x−h=±kx - h = \pm\sqrt{k}, so x=h±kx = h \pm \sqrt{k}. Get the square alone first.

Worked example

Solve (x−3)2=25(x - 3)^2 = 25.

x−3=5x - 3 = 5 or x−3=−5x - 3 = -5
Square roots.
x=8x = 8 or x=−2x = -2
Add 3.

Your turn, guided

Your turn: the positive solution of 2x2=722x^2 = 72.

66
x² = 36.

Practice ★

  1. Solve x2=100x^2 = 100. Enter all solutions.
  2. Solve x2=25x^2 = 25. Enter all solutions.
  1. Solve 3x2−3=03x^2 - 3 = 0. Enter all solutions.

Watch out!

Practice ★★

  1. Solve (x−7)2=64(x - 7)^2 = 64. Enter all solutions.
  2. Solve (x−8)2=1(x - 8)^2 = 1. Enter all solutions.

Exit ticket

Three questions to check you have got it.

  1. Solve x2=100x^2 = 100. Enter all solutions.
  2. Solve x2=25x^2 = 25. Enter all solutions.
  3. Solve x2=36x^2 = 36. Enter all solutions.

Summary card

Solving by square roots

Isolate the square; then ±  \pm\sqrt{\;}.

Watch out: two solutions, not one.

2. The zero product property

Warm-up

Three quick questions from earlier lessons.

Discover

a⋅b=0a \cdot b = 0. What must be true?

  1. a=0a = 0 or b=0b = 0
  2. Both are 0

Key idea

The zero product property.

If A⋅B=0A \cdot B = 0, then A=0A = 0 or B=0B = 0. So to solve a quadratic equation, write it as (factor)(factor) = 0 and set each factor equal to zero. This only works with 0 on one side.

Worked example

Solve (3x−6)(x+4)=0(3x - 6)(x + 4) = 0.

3x−6=03x - 6 = 0, x=2x = 2
First factor.
x+4=0x + 4 = 0, x=−4x = -4
Second factor.

Your turn, guided

Your turn: the non-zero solution of x(x−9)=0x(x - 9) = 0.

99
x − 9 = 0.

Practice ★

  1. Solve x(x−3)=0x(x - 3) = 0. Enter all solutions.
  2. Solve x(x−2)=0x(x - 2) = 0. Enter all solutions.
  1. Solve (4x−6)(x+8)=0(4x - 6)(x + 8) = 0. Enter all solutions.

Watch out!

Practice ★★

  1. To solve (x−3)(x+7)=4(x - 3)(x + 7) = 4, Ana writes “x−3=4x - 3 = 4 or x+7=4x + 7 = 4”. Is that right?
    1. No: the zero product property only works when the product is 0
    2. Yes: set each factor equal to 4
    3. Only the first factor can equal 4
  2. To solve (x−7)(x+6)=17(x - 7)(x + 6) = 17, Ana writes “x−7=17x - 7 = 17 or x+6=17x + 6 = 17”. Is that right?
    1. No: the zero product property only works when the product is 0
    2. Yes: set each factor equal to 17
    3. Only the first factor can equal 17

Exit ticket

Three questions to check you have got it.

  1. Solve (4x−6)(x+8)=0(4x - 6)(x + 8) = 0. Enter all solutions.
  2. Solve (2x−5)(x+9)=0(2x - 5)(x + 9) = 0. Enter all solutions.
  3. Solve (5x−3)(x+8)=0(5x - 3)(x + 8) = 0. Enter all solutions.

Summary card

Zero product property

Product = 0: set each factor to 0.

Watch out: never divide by xx; you lose x=0x = 0.

3. Solve quadratic equations by factoring

Warm-up

Three quick questions from earlier lessons.

Discover

To factor x2+7x+10x^2 + 7x + 10, you need two numbers that multiply to 10 and add to 7. Which?

  1. 2 and 5
  2. 1 and 10

Key idea

Factor, then use the zero product property.

x2+bx+c=(x+m)(x+n)x^2 + bx + c = (x + m)(x + n) when m⋅n=cm \cdot n = c and m+n=bm + n = b. To solve by factoring: bring everything to one side, factor, set each factor to 0. A negative cc means one number is negative.

Worked example

Solve x2−2x−15=0x^2 - 2x - 15 = 0.

(x−5)(x+3)=0(x - 5)(x + 3) = 0
−5 · 3 = −15, −5 + 3 = −2.
x=5x = 5 or x=−3x = -3
Zero product.

Your turn, guided

Your turn: x2+9x+14=(x+2)(x+  ?)x^2 + 9x + 14 = (x + 2)(x + \;?).

77
2 · 7 = 14.

Practice ★

  1. x2+8x+15=(x+3)(x+  ?)x^2 + 8x + 15 = (x + 3)(x + \;?)
  2. x2+17x+72=(x+9)(x+  ?)x^2 + 17x + 72 = (x + 9)(x + \;?)
  1. Solve x2−9x+14=0x^2 - 9x + 14 = 0 by factoring. Enter all solutions.

Watch out!

Practice ★★

  1. Solve x2+8x−9=0x^2 + 8x - 9 = 0 by factoring. Enter all solutions.
  2. Solve x2+7x−8=0x^2 + 7x - 8 = 0 by factoring. Enter all solutions.

Exit ticket

Three questions to check you have got it.

  1. Solve x2−9x+14=0x^2 - 9x + 14 = 0 by factoring. Enter all solutions.
  2. Solve x2−17x+72=0x^2 - 17x + 72 = 0 by factoring. Enter all solutions.
  3. Solve x2−16x+63=0x^2 - 16x + 63 = 0 by factoring. Enter all solutions.

Summary card

Solving by factoring

Multiply to cc, add to bb; then each factor =0= 0.

Watch out: the factor x−4x - 4 gives x=4x = 4.

4. Completing the square

Warm-up

Three quick questions from earlier lessons.

Discover

x2+6x+9=(x+3)2x^2 + 6x + 9 = (x + 3)^2. Where does the 9 come from?

Not to scale
  1. Step 1x² + 6x: a square and a strip.
  2. Step 2Split the strip into two 3x strips and put them on two sides.
  3. Step 3The corner is missing: add (6 ÷ 2)² = 9 to get (x + 3)².
  1. (6÷2)2(6 \div 2)^2
  2. 6+36 + 3

Key idea

Complete the square.

x2+bx+(b2)2=(x+b2)2x^2 + bx + \left(\tfrac{b}{2}\right)^2 = \left(x + \tfrac{b}{2}\right)^2. To solve x2+bx=kx^2 + bx = k: add (b2)2\left(\tfrac{b}{2}\right)^2 to both sides, write the left side as a square, then take ±\pm square roots. The same move turns x2+bx+cx^2 + bx + c into vertex form.

Worked example

Solve x2+4x=1x^2 + 4x = 1.

x2+4x+4=5x^2 + 4x + 4 = 5
Add (4/2)² = 4 to both sides.
(x+2)2=5(x + 2)^2 = 5
Square.
x=−2±5x = -2 \pm \sqrt{5}
±√5, subtract 2.

Your turn, guided

Your turn: the number that completes x2+10x+  ?x^2 + 10x + \;?.

2525
(10/2)².

Practice ★

  1. What number makes x2+10x+  ?x^2 + 10x + \;? a perfect square?
  2. What number makes x2+18x+  ?x^2 + 18x + \;? a perfect square?
  1. x2+12x+22=(x+6)2+  ?x^2 + 12x + 22 = (x + 6)^2 + \;?

Watch out!

Practice ★★

  1. Solve x2+6x=8x^2 + 6x = 8 by completing the square. Enter all solutions (use sqrt).
  2. Solve x2+2x=9x^2 + 2x = 9 by completing the square. Enter all solutions (use sqrt).

Exit ticket

Three questions to check you have got it.

  1. What number makes x2+10x+  ?x^2 + 10x + \;? a perfect square?
  2. What number makes x2+18x+  ?x^2 + 18x + \;? a perfect square?
  3. What number makes x2+4x+  ?x^2 + 4x + \;? a perfect square?

Summary card

Completing the square

Add (b/2)2(b/2)^2 to both sides; then square roots.

Watch out: halve, then square; and do it on both sides.

5. The quadratic formula and the discriminant

Warm-up

Three quick questions from earlier lessons.

Discover

x2+2x+5=0x^2 + 2x + 5 = 0. Completing the square gives (x+1)2=−4(x + 1)^2 = -4. How many real solutions?

  1. None
  2. Two

Key idea

The quadratic formula solves every quadratic equation.

The solutions of ax2+bx+c=0ax^2 + bx + c = 0 are x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}. The discriminant b2−4acb^2 - 4ac tells how many real solutions: positive, two; zero, one; negative, none.

Worked example

Solve 2x2+3x−2=02x^2 + 3x - 2 = 0.

b2−4ac=9+16=25b^2 - 4ac = 9 + 16 = 25
a = 2, b = 3, c = −2.
x=−3±54x = \dfrac{-3 \pm 5}{4}
Formula.
x=12x = \tfrac{1}{2} or x=−2x = -2
Two solutions.

Your turn, guided

Your turn: the discriminant of x2+5x+6=0x^2 + 5x + 6 = 0.

11
25 − 24.

Practice ★

  1. Find the discriminant b2−4acb^2 - 4ac of 1x2+12x+2=01x^2 + 12x + 2 = 0.
  2. Find the discriminant b2−4acb^2 - 4ac of 1x2+11x+9=01x^2 + 11x + 9 = 0.
  1. How many real solutions has x2+7x+49=0x^2 + 7x + 49 = 0?
    1. None
    2. Two
    3. One

Watch out!

Practice ★★

  1. Solve 2x2+5x−1=02x^2 + 5x - 1 = 0 with the quadratic formula. Enter all solutions (use sqrt).
  2. Solve 2x2+7x−6=02x^2 + 7x - 6 = 0 with the quadratic formula. Enter all solutions (use sqrt).
  1. Boss problem. A rectangular garden is 7 m longer than it is wide, and its area is 294 m². Let ww be the width in meters.

Exit ticket

Three questions to check you have got it.

  1. Find the discriminant b2−4acb^2 - 4ac of 1x2+12x+2=01x^2 + 12x + 2 = 0.
  2. Find the discriminant b2−4acb^2 - 4ac of 1x2+11x+9=01x^2 + 11x + 9 = 0.
  3. Find the discriminant b2−4acb^2 - 4ac of 2x2+8x+5=02x^2 + 8x + 5 = 0.

Summary card

The quadratic formula

x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}; the discriminant counts the solutions.

Watch out: 2a2a divides the whole numerator.

6. Modeling with quadratic equations

Warm-up

Three quick questions from earlier lessons.

Discover

A square has an area of 49 cm². Which equation gives its side s?

  1. s² = 49
  2. 4s = 49

Key idea

Write the equation from the situation, solve it, then check the context.

Name the unknown, translate the words into an equation (often a product: area = length × width, or a formula such as h=−5t2+vth = -5t^2 + vt), solve it, and keep only the solutions that make sense: lengths and times are not negative, and the launch time is not the landing time.

Worked example

A rectangle is 3 m longer than wide, with area 40 m².

w(w+3)=40w(w + 3) = 40
Equation.
w2+3w−40=0w^2 + 3w - 40 = 0, (w+8)(w−5)=0(w + 8)(w - 5) = 0
Solve.
w=5w = 5 m (reject −8-8)
Context.

Your turn, guided

Your turn: a square of area 81 m². Its side, in meters?

99
s² = 81, s > 0.

Practice ★

  1. The product of two consecutive whole numbers is 240. Which equation models this, with x the smaller number?
    1. x(x + 1) = 240
    2. x + (x + 1) = 240
    3. x² + 1 = 240
  2. The product of two consecutive whole numbers is 210. Which equation models this, with x the smaller number?
    1. x(x + 1) = 210
    2. x + (x + 1) = 210
    3. x² + 1 = 210
  1. A rectangular garden is 6 m longer than it is wide. Its area is 216 m². Write and solve an equation for the width w. What is the width, in meters?

Watch out!

Practice ★★

  1. A ball is kicked upward. Its height after t seconds is h = −5t² + 35t meters. After how many seconds does it land?
  2. A ball is kicked upward. Its height after t seconds is h = −5t² + 30t meters. After how many seconds does it land?
  1. Boss problem. A photo 22 cm by 14 cm gets a frame of the same width x cm on every side. The framed photo covers 660 cm².

Exit ticket

Three questions to check you have got it.

  1. A rectangular garden is 6 m longer than it is wide. Its area is 216 m². Write and solve an equation for the width w. What is the width, in meters?
  2. A rectangular garden is 3 m longer than it is wide. Its area is 88 m². Write and solve an equation for the width w. What is the width, in meters?
  3. A rectangular garden is 7 m longer than it is wide. Its area is 144 m². Write and solve an equation for the width w. What is the width, in meters?

Summary card

Modeling with quadratic equations

Translate, solve, then keep the solutions that fit the situation.

Watch out: a negative length or the start time is not the answer.

7. Rational and irrational numbers; repeating decimals

Warm-up

Three quick questions from earlier lessons.

Discover

13=0.333…\tfrac{1}{3} = 0.333\ldots goes on forever. Is it rational?

  1. Yes
  2. No: it never ends

Key idea

Rational numbers are fractions; their decimals end or repeat.

A rational number can be written ab\tfrac{a}{b} with integers aa, b≠0b \neq 0; its decimal expansion terminates or repeats. An irrational number cannot: its decimal never ends and never repeats. n\sqrt{n} is irrational unless nn is a perfect square; π\pi is irrational.

Worked example

Write 0.27‾0.\overline{27} as a fraction.

100x=27.2727…100x = 27.2727\ldots
Let x = 0.2727…
99x=2799x = 27
Subtract x.
x=2799=311x = \tfrac{27}{99} = \tfrac{3}{11}
Simplify.

Your turn, guided

Your turn: write 0.5‾0.\overline{5} as a fraction.

59\tfrac{5}{9}
10x − x = 5.

Practice ★

  1. Which number is irrational?
    1. 2\sqrt{ 2 }
    2. 4\sqrt{ 4 }
    3. 0.1‾0.\overline{ 1 }
  2. Which number is irrational?
    1. 10\sqrt{ 10 }
    2. 16\sqrt{ 16 }
    3. 0.3‾0.\overline{ 3 }
  1. Write 0.1‾0.\overline{ 1 } (the digit 1 repeating forever) as a fraction.

Watch out!

Practice ★★

  1. Write 0.48‾0.\overline{ 48 } (the digits 48 repeating forever) as a fraction.
  2. Write 0.70‾0.\overline{ 70 } (the digits 70 repeating forever) as a fraction.

Exit ticket

Three questions to check you have got it.

  1. Write 0.1‾0.\overline{ 1 } (the digit 1 repeating forever) as a fraction.
  2. Write 0.2‾0.\overline{ 2 } (the digit 2 repeating forever) as a fraction.
  3. Write 0.8‾0.\overline{ 8 } (the digit 8 repeating forever) as a fraction.

Summary card

Rational and irrational

Ends or repeats: rational. Neither: irrational.

Watch out: a repeating decimal is a fraction over 9, 99, …

8. Systems with a line and a conic

Warm-up

Three quick questions from earlier lessons.

Discover

How many points can a line and a circle have in common?

  1. 0, 1 or 2
  2. Always 2

Key idea

To find where a line meets a conic, solve the two equations together.

Substitute the line into the conic: you get a quadratic in one variable. Its real solutions give the intersection points: 2 solutions, 2 points; a double root, a tangent; no real root, no intersection. Then find the other coordinate from the line.

Worked example

Where does y=x+1y = x + 1 meet x2+y2=25x^2 + y^2 = 25?

x2+(x+1)2=25x^2 + (x + 1)^2 = 25
Substitute y.
2x2+2x−24=02x^2 + 2x - 24 = 0, so x2+x−12=0x^2 + x - 12 = 0
Expand and simplify.
(3,4)(3, 4) and (−4,−3)(-4, -3)
(x + 4)(x − 3) = 0, then y = x + 1.

Your turn, guided

Your turn: where does y=3y = 3 meet x2+y2=25x^2 + y^2 = 25?

x2=16x^2 = 16
Substitute y = 3.
x=4x = 4 or x=−4x = -4
Two square roots: type the negative one.

Practice ★

  1. The circle x2+y2=169x^2 + y^2 = 169 and the line y=12y = 12 meet at two points. Give their xx-coordinates.
  2. The circle x2+y2=25x^2 + y^2 = 25 and the line y=4y = 4 meet at two points. Give their xx-coordinates.
  3. The circle x2+y2=400x^2 + y^2 = 400 and the line y=16y = 16 meet at two points. Give their xx-coordinates.

Watch out!

Practice ★★

  1. How many points do the circle x2+y2=16x^2 + y^2 = 16 and the line y=11y = 11 have in common?
    1. None
    2. One
    3. Two
  1. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+6y = x + 6.
  2. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+20y = x + 20.
  3. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+2y = x + 2.

Exit ticket

Three questions to check you have got it.

  1. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+6y = x + 6.
  2. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+20y = x + 20.
  3. Find the xx-coordinates of the points where y=x2y = x^2 meets y=x+2y = x + 2.

Summary card

A line and a conic

  1. Substitute the line into the conic.
  2. Solve the quadratic: 0, 1 or 2 real solutions.
  3. Find the other coordinate from the line.

Watch out: a negative square means no intersection.

Answers: Quadratic equations

1. Solve quadratic equations by taking square roots

A

  1. 10, −1010,\ -10
  2. 5, −55,\ -5

B

  1. 1, −11,\ -1

C

  1. 15, −115,\ -1
  2. 9, 79,\ 7

D

  1. 10, −1010,\ -10
  2. 5, −55,\ -5
  3. 6, −66,\ -6

2. The zero product property

A

  1. 0, 30,\ 3
  2. 0, 20,\ 2

B

  1. 32, −8\frac{3}{2},\ -8

C

  1. No: the zero product property only works when the product is 0
  2. No: the zero product property only works when the product is 0

D

  1. 32, −8\frac{3}{2},\ -8
  2. 52, −9\frac{5}{2},\ -9
  3. 35, −8\frac{3}{5},\ -8

3. Solve quadratic equations by factoring

A

  1. 55
  2. 88

B

  1. 7, 27,\ 2

C

  1. −9, 1-9,\ 1
  2. −8, 1-8,\ 1

D

  1. 7, 27,\ 2
  2. 8, 98,\ 9
  3. 9, 79,\ 7

4. Completing the square

A

  1. 2525
  2. 8181

B

  1. −14-14

C

  1. −3+17, −17−3-3 + \sqrt{17},\ - \sqrt{17} - 3
  2. −1+10, −10−1-1 + \sqrt{10},\ - \sqrt{10} - 1

D

  1. 2525
  2. 8181
  3. 44

5. The quadratic formula and the discriminant

A

  1. 136136
  2. 8585

B

  1. None

C

  1. −54+334, −334−54- \frac{5}{4} + \frac{\sqrt{33}}{4},\ - \frac{\sqrt{33}}{4} - \frac{5}{4}
  2. −74+974, −974−74- \frac{7}{4} + \frac{\sqrt{97}}{4},\ - \frac{\sqrt{97}}{4} - \frac{7}{4}

D

  1. 2121

E

  1. 136136
  2. 8585
  3. 2424

6. Modeling with quadratic equations

A

  1. x(x + 1) = 240
  2. x(x + 1) = 210

B

  1. 1212

C

  1. 77
  2. 66

D

  1. 352352

E

  1. 1212
  2. 88
  3. 99

7. Rational and irrational numbers; repeating decimals

A

  1. 2\sqrt{ 2 }
  2. 10\sqrt{ 10 }

B

  1. 19\frac{1}{9}

C

  1. 1633\frac{16}{33}
  2. 7099\frac{70}{99}

D

  1. 19\frac{1}{9}
  2. 29\frac{2}{9}
  3. 89\frac{8}{9}

8. Systems with a line and a conic

A

  1. 5, −55,\ -5
  2. 3, −33,\ -3
  3. 12, −1212,\ -12

B

  1. None

C

  1. 3, −23,\ -2
  2. 5, −45,\ -4
  3. 2, −12,\ -1

D

  1. 3, −23,\ -2
  2. 5, −45,\ -4
  3. 2, −12,\ -1