Porisma

Algebra readiness

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By the end, you can

  • Refresh equations, powers and logarithms
  • Complete the square and use the factor theorem
  • Recognise partial fractions, trigonometry and basic calculus

about an hour for the lesson · practise at your own pace.

The central idea

Equivalent algebraic forms reveal different information. Factoring finds zeros, completing the square exposes a quadratic vertex, and logarithms turn multiplication into addition. Keep the number system and domain in view: denominators cannot vanish and real logarithms need positive arguments.

Six discriminant and leading-coefficient cases

How roots and opening fit together

D = 4: two distinct real roots. The curve crosses the axis at −1 and 1.

Loading graph…

Read a quadratic in two checks

  1. Check the discriminant.
  2. Count the real roots.
  3. Use the opening to decide signs.
Six schematic parabolas: for D>0 both upward and downward curves cross the x-axis twice; for D=0 each touches it at its vertex; for D<0 the upward curve stays above and the downward curve below. D=b²−4ac. No coordinates are prescribed by the source.
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Exponential and logarithm graphs for both base ranges

Compare exponentials and logarithms

Domain: all real inputs. Range: y > 0. The curve increases; y = 0 is a horizontal asymptote, never an attained value.

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Choose an exponential or logarithm method

  1. Check the base and domain.
  2. Rewrite using a common base or logarithms.
  3. Check answers in the original equation.
Four source graph cases. With a>1 the exponential increases through (0,1) and logarithm increases through (1,0); with 0<a<1 each decreases through the same respective intercept. Exponentials stay above the x-axis; logarithms have domain x>0 and vertical asymptote x=0. Curves are schematic.
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Circular measure and right-triangle ratios

Illustration: r = AC = 3 units.

Acute angle θ45° = π/4 rad
Try an example

Arc length

s=rθs=r\theta

θ in radians
OPQrθs

The highlighted arc PQ subtends θ at centre O.

s ≈ 2.3562 units.

Sector area

A=12r2θA=\tfrac12r^2\theta

θ in radians
OPQrθ

The shaded region is sector POQ, with area A.

A ≈ 3.5343 square units.

Right-triangle ratios

The right angle is at B; the acute angle θ is at C. AB is opposite, BC adjacent and AC the hypotenuse.

ABCABBCACθ

AB ≈ 2.1213, BC ≈ 2.1213, AC = 3 units.

sin⁡θ=ABAC\sin\theta=\dfrac{AB}{AC}

≈ 0.7071

cos⁡θ=BCAC\cos\theta=\dfrac{BC}{AC}

≈ 0.7071

tan⁡θ=ABBC\tan\theta=\dfrac{AB}{BC}

≈ 1

Compare with the original three panels

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Three source panels: circle with centre O, radius r, angle theta and arc PQ length s; separate circle with sector POQ area A; right triangle ABC with right angle B and acute theta at C, opposite AB, adjacent BC and hypotenuse AC.
Three source panels: circle with centre O, radius r, angle theta and arc PQ length s; separate circle with sector POQ area A; right triangle ABC with right angle B and acute theta at C, opposite AB, adjacent BC and hypotenuse AC.
Three source panels: circle with centre O, radius r, angle theta and arc PQ length s; separate circle with sector POQ area A; right triangle ABC with right angle B and acute theta at C, opposite AB, adjacent BC and hypotenuse AC.
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General triangle side and angle names

Match a lowercase side with the opposite uppercase angle. These rules apply to general triangles; no right angle is assumed.

ABCabc
Each side is opposite its matching angle. The sketch supplies no side lengths.

Match opposite sides and angles

Opposite angle A

a=BCa=BC

Opposite angle B

b=CAb=CA

Opposite angle C

c=ABc=AB

Cosine rule: use the included angle

Use two sides and their included angle, or use all three sides to find an angle. Choose the form with the required side opposite its matching angle.

For side a, opposite A

a2=b2+c2−2bccos⁡A.\begin{aligned}a^2&=b^2+c^2\\&\quad-2bc\cos A.\end{aligned}

For side b, opposite B

b2=a2+c2−2accos⁡B.\begin{aligned}b^2&=a^2+c^2\\&\quad-2ac\cos B.\end{aligned}

For side c, opposite C

c2=a2+b2−2abcos⁡C.\begin{aligned}c^2&=a^2+b^2\\&\quad-2ab\cos C.\end{aligned}

Try the included-angle example

Sine rule: keep each opposite pair together

sin⁡Aa=sin⁡Bb=sin⁡Cc.\begin{aligned}\frac{\sin A}{a}&=\frac{\sin B}{b}\\&=\frac{\sin C}{c}.\end{aligned}

If inverse sine gives an angle, check whether its supplementary angle also satisfies the triangle's other data.

Compare with the original naming sketch

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General triangle ABC has angle labels A,B,C and opposite sides a=BC,b=CA,c=AB. No right angle is assumed.
General triangle ABC has angle labels A,B,C and opposite sides a=BC,b=CA,c=AB. No right angle is assumed.
General triangle ABC has angle labels A,B,C and opposite sides a=BC,b=CA,c=AB. No right angle is assumed.
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Sine cosine and tangent graphs

Read one trigonometric graph at a time

Domain: all real inputs. Range: [−1, 1]. Period: 2π. Sine is odd: sin(−x) = −sin x.

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Read a trigonometric graph

  1. Choose the function and interval.
  2. Mark intercepts and excluded poles.
  3. Use the period to repeat the pattern.
Three source graph panels over minus 2pi to 2pi: sine and cosine range from minus 1 to 1 with period 2pi; sine is odd and cosine even. Tangent is odd with periodpi, separate increasing branches, and excluded vertical asymptotes at odd multiples of pi/2.
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Principal inverse-trigonometric graphs

See the principal angle range

Domain: [−1, 1]. Principal range: [−π/2, π/2]. Increasing, with both endpoints included.

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Choose a principal inverse value

  1. Check the inverse function domain.
  2. Find the angle in its principal range.
  3. Check by applying the original function.
Principal inverse sine increases on closed domain[-1,1] from minuspi/2 to pi/2. Inverse cosine decreases on[-1,1] from pi to 0, passing through (0, pi/2). Inverse tangent increases for every real input, with horizontal asymptotes minuspi/2 and pi/2 that are never attained.
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Explore

Use the worked example below, then change one input or assumption. Predict which part of the answer must change before calculating.

Related interactive example

Explore this related example, then return to the prediction task above.

Where is x(x − 5)/(x + 3) negative?

Drag the test value. Each factor has a sign; the product's sign decides the inequality.

x(x−5)x+3<0\frac{x(x-5)}{x+3}<0

Choose an interval to test it. Shading marks the solution intervals.

Intervalxx − 5x + 3product
negativenegativenegativenegative
negativenegativepositivepositive
positivenegativepositivenegative
positivepositivepositivepositive
Test value of xx = 2.00
Try an example

(x)+(x − 5)−(x + 3)+⇒ product negative

Is it < 0?
Yes
x is in the solution set

The shaded strips are where the expression satisfies the inequality; they match the green-tinted columns of the chart.

  • Expression, and the x-values where it is negative
  • Your test value
Key idea

The product changes sign at each simple root. With three simple roots, the sign alternates, and the answer is the negative intervals.

Ideas to try
  1. Test x=−4x=-4, −1-1, 22 and 66. Which are solutions?
  2. Move the test value across x=−3x=-3: what happens to the denominator's sign?
Open the full interactive problem →

Worked example

Solve x2−6x+5=0x^2-6x+5=0. Completing the square gives (x−3)2=4(x-3)^2=4, hence x=1x=1 or x=5x=5. The same expression is (x−1)(x−5)(x-1)(x-5); its vertex form also shows that its minimum is −4-4 at x=3x=3.

Squaring can introduce candidates that do not satisfy the original equation.

Check your understanding

Solve x+2=x\sqrt{x+2}=x over the real numbers.

Practice questions

Practice section 1 · Questions 1–6

Complete the square with and without a leading coefficient · Solve a reciprocal equation using the quadratic formula · +4 more

Practice section 2 · Questions 7–12

Apply fractional and negative index laws · Use a positive exponential substitution · +4 more

Practice section 3 · Questions 13–18

Include both powers of a repeated linear factor · Combine a linear factor with an irreducible quadratic · +4 more

Practice section 4 · Questions 19–24

Control the square-root sign in a trigonometric substitution · Prove the triple-angle cosine identity · +4 more

Practice section 5 · Questions 25–30

Use the requested product rule · Keep the quotient-rule numerator order · +4 more

Practice section 6 · Questions 31–36

Complete the square · Find exact quadratic roots · +4 more

Practice section 7 · Questions 37–42

Simplify fractional indices · Solve with a common exponential base · +4 more

Practice section 8 · Questions 43–47

Prove a fourth-power trigonometric identity · Combine sine and cosine to prove bounds · +3 more

More practice

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