Learning Objectives
- Solve quadratic equations by completing the square and by the quadratic formula.
- Give exact answers in surd form when appropriate.
- Choose a suitable method for a particular quadratic equation.
- Form quadratic equations from geometric and numerical problems.
- Solve equations containing numerical or linear algebraic denominators and check excluded values.
Key Terms
- Quadratic formula
- A general formula for solving ax^2+bx+c=0.
- Exact value
- A value written without rounding, often involving a fraction or surd.
- Fractional equation
- An equation containing one or more fractions, possibly with the variable in a denominator.
- Extraneous solution
- A value produced by algebraic manipulation that does not satisfy the original equation.
- Lowest common multiple
- The smallest expression divisible by every denominator.
- Rearrange
- Rewrite an equation in an equivalent form.
- Verification
- Substitution of a proposed solution into the original equation to confirm it works.
What This Chapter Covers
- Completing the square to solve equations
- Using the quadratic formula
- Exact surd solutions and sensible rounding
- Constructing quadratics from contexts
- Clearing denominators in fractional equations
Choosing A Method
Factorisation is usually fastest when the factors are obvious. Completing the square is especially useful when a question asks for that form or when a graph’s turning point is required. The quadratic formula works for every quadratic equation and is essential when factorisation is difficult.
| Method | Best Used When | Main Risk |
|---|---|---|
| Factorisation | Integer or simple rational factors are visible | Missing a factor or one of the two roots |
| Completing the square | Turning point or exact structural form is useful | Sign errors when compensating for the square |
| Quadratic formula | No easy factorisation exists | Substituting signs incorrectly, especially b<0 |
| Graphical method | An approximate solution is required from a graph | Reading beyond the accuracy of the scale |
Solving By Completing The Square
Rearrange to x^2+bx+c=0 when the leading coefficient is 1. Write the quadratic part as a square, isolate the square and take both positive and negative square roots.
Worked Example: Complete The Square
Question: Solve x^2-6x+1=0.
- Write x^2-6x+1=(x-3)^2-8.
- Set equal to zero: (x-3)^2=8.
- Take both roots: x-3=\pm\sqrt8=\pm2\sqrt2.
Answer: x=3\pm2\sqrt2.
If the leading coefficient is not 1, divide through first when convenient, or factor the coefficient from the quadratic and linear terms before completing the square.
Worked Example: Leading Coefficient Not One
Question: Solve 2x^2+8x-3=0 by completing the square.
- Divide by 2: x^2+4x-\frac32=0.
- Write (x+2)^2-4-\frac32=0.
- (x+2)^2=\frac{11}{2}.
- Take both square roots.
Answer: x=-2\pm\sqrt{\frac{11}{2}}=-2\pm\frac{\sqrt{22}}{2}.
The Quadratic Formula
Use this only after identifying a, b and c from ax^2+bx+c=0.
Substitution Checklist
- Rearrange the equation into standard form with zero on one side.
- Write a, b and c, including their signs.
- Substitute using brackets, especially for a negative b or c.
- Calculate the expression under the square root before taking the root.
- Use both the plus and minus signs.
- Give an exact form or round only at the final step as directed.
Worked Example: Formula With A Negative Coefficient
Question: Solve 3x^2-4x-2=0.
- a=3,\ b=-4,\ c=-2.
- Substitute: x=\frac{-(-4)\pm\sqrt{(-4)^2-4(3)(-2)}}{2(3)}.
- Simplify the discriminating expression: 16+24=40.
- \sqrt{40}=2\sqrt{10}.
Answer: x=\frac{2\pm\sqrt{10}}{3}.
Exact Answers And Decimal Answers
If a question says “give your answer in exact form”, leave surds and fractions unrounded. If a context requires a measurement, a decimal answer may be appropriate. Keep all calculator digits during working and round only the final result. If a length is requested, reject any negative root that is not meaningful in context.
Constructing A Quadratic Equation
Translate the information into algebra, form an equation, then solve. Common contexts involve products of consecutive numbers, rectangles, areas and speed-time relationships. Define the variable clearly before forming the equation.
Worked Example: Rectangle Dimensions
Question: A rectangle is 4 cm longer than it is wide and has area 96 cm². Find its dimensions.
- Let the width be x cm, so the length is x+4 cm.
- Area equation: x(x+4)=96.
- Rearrange: x^2+4x-96=0.
- Factorise: (x+12)(x-8)=0.
- Reject x=-12 because a width cannot be negative.
Answer: Width 8 cm and length 12 cm.
Worked Example: Consecutive Integers
Question: The product of two consecutive positive integers is 156. Find the integers.
- Let the smaller integer be n; the next is n+1.
- n(n+1)=156, so n^2+n-156=0.
- Factorise: (n-12)(n+13)=0.
- Use the positive solution.
Answer: The integers are 12 and 13.
Fractional Equations
A fractional equation may contain numerical denominators, algebraic denominators or both. First state any values that make a denominator zero. Then multiply every term by the lowest common denominator. This clears the fractions and produces a linear or quadratic equation.
Clearing Denominators Safely
- Factor denominators if necessary and identify excluded values.
- Find the lowest common denominator.
- Multiply every term on both sides by that denominator.
- Simplify before expanding unnecessarily.
- Solve the resulting equation.
- Check each solution in the original equation and reject excluded or invalid values.
Worked Example: Linear Fractional Equation
Question: Solve \frac{x}{x+2}=3.
- Restriction: x\ne-2.
- Multiply by x+2: x=3(x+2).
- x=3x+6, so -2x=6.
Answer: x=-3.
Worked Example: Two Algebraic Denominators
Question: Solve \frac{2}{x-1}+\frac{1}{x+1}=1.
- Restrictions: x\ne1,-1.
- Multiply by (x-1)(x+1).
- 2(x+1)+(x-1)=x^2-1.
- Simplify: 3x+1=x^2-1, hence x^2-3x-2=0.
- Use the quadratic formula.
Answer: x=\frac{3\pm\sqrt{17}}{2}; neither value is excluded.
Worked Example: Equation Involving A Reciprocal
Question: Solve x+\frac{6}{x}=5.
- Restriction: x\ne0.
- Multiply through by x: x^2+6=5x.
- Rearrange: x^2-5x+6=0.
- Factorise: (x-2)(x-3)=0.
Answer: x=2 or x=3.
Checking For Invalid Solutions
Multiplying by an expression that can equal zero may hide the original restriction. Always substitute answers into the original equation. In more advanced equations, squaring both sides can also create an extraneous solution; verification protects against this even when the question does not explicitly request it.
Forming And Solving In Context
After finding algebraic roots, return to the wording of the problem. Units, positivity, integer conditions and geometric constraints determine which roots are acceptable. State why a rejected root is unsuitable rather than simply deleting it.
Examination Guidance
- Write the quadratic in standard form before using the formula.
- Identify a, b and c with signs explicitly.
- For fractional equations, multiply every term by the common denominator, not only the fractional terms.
- Keep exact surd forms unless the question requests a decimal.
- In word problems, define the variable and state why any root is rejected.
Common Mistakes
- Using -b as b when b is already negative.
- Forgetting the \pm when taking a square root.
- Rounding the square root too early and losing accuracy.
- Cancelling across an equation instead of multiplying every term by the common denominator.
- Accepting a value that makes an original denominator zero.
Knowledge Check And Practice
1. Solve x^2+4x-7=0 by completing the square.
2. Solve 5x^2+2x-3=0.
3. Solve 2x^2-3x-7=0 using the formula.
4. The area of a square is 196 cm². Form and solve an equation for its side length.
5. Solve \frac{x+1}{x-2}=2.
6. Solve \frac{3}{x}+1=\frac{5}{x}.
7. Solve x-\frac{12}{x}=1.
8. Why should answers to fractional equations be checked in the original equation?
