Solving Quadratic Equations
This topic covers the essential algebraic techniques for solving quadratic equations, which model parabolic relationships common in physics and engineering. Mastery of these methods is crucial for finding unknown variables without a calculator.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- A quadratic equation must first be arranged into the standard form `ax2 + bx + c = 0`, where `a` is not zero.
- The solutions to a quadratic equation, also known as its roots, correspond to the x-intercepts of its graph.
- There are three main algebraic methods: factorising (quickest for simple cases), completing the square (also useful for finding a parabola's vertex), and the quadratic formula (works for all cases).
- A quadratic equation can have two distinct real roots, one repeated real root, or no real roots.
- Be alert for 'disguised' quadratics, such as `ax4 + bx2 + c = 0`, which can be solved by substituting a new variable like `y = x2`.
- Approximate solutions can be found graphically by identifying where the curve `y = ax2 + bx + c` crosses the x-axis.
Diagram
Formulae
x = (-b ± √(b2 - 4ac)) / (2a) This is the quadratic formula. Use it when factorising is not obvious or when the roots are not rational numbers. You must memorise this formula.
x2 + kx ⇒ (x + k/2)2 - (k/2)2 This is the core transformation for completing the square. It rewrites the `x` terms as a perfect square, which is useful for solving the equation or finding the vertex of the parabola.
Definitions
- Quadratic Equation
- A second-degree polynomial equation of the form `ax2 + bx + c = 0`, where `a`, `b`, and `c` are constants and `a ≠ 0`.
- Roots
- The solutions to a quadratic equation; the values of `x` for which the equation is true.
Worked example
Solve for all real values of `p`: `3 - 5/p = 2/p2`.
- 1
First, state the constraint that `p` cannot be zero, as it appears in the denominator.
- 2
To eliminate the fractions, multiply every term in the equation by `p2`:
`3(p2) - (5/p)(p2) = (2/p2)(p2)`.
- 3
Simplify the equation:
`3p2 - 5p = 2` - 4
Rearrange into the standard quadratic form `ax2 + bx + c = 0`:
`3p2 - 5p - 2 = 0`.
- 5
Attempt to factorise.
We need two numbers that multiply to `3 × -2 = -6` and add to `-5`.
These numbers are `-6` and `+1`.
- 6
Split the middle term:
`3p2 - 6p + p - 2 = 0`.
- 7
Factor by grouping:
`3p(p - 2) + 1(p - 2) = 0`.
- 8
Complete the factorisation:
`(3p + 1)(p - 2) = 0`.
- 9
Solve for the two possible roots:
`3p + 1 = 0` gives `p = -1/3`, and `p - 2 = 0` gives `p = 2`
Answer: p = 2 or p = -1/3
Common mistakes
- ×Sign errors are the most frequent mistake. When using the formula `x = (-b ± √(b2 - 4ac)) / (2a)`, be extremely careful if `b` or `c` are negative. For instance, if `b = -5`, then `-b` is `+5`.
- ×When solving an equation like `(x-3)2 = 16`, remember to take both positive and negative square roots: `x-3 = 4` AND `x-3 = -4`, giving two distinct solutions.
- ×Forgetting to rearrange the equation to equal zero (`⋯ = 0`) before applying the quadratic formula or attempting to factorise.
- ×In 'disguised' quadratics involving fractions, failing to check if a solution violates an initial constraint (e.g., division by zero).
No-calculator tips
- ✓Before using the quadratic formula, always check if the expression can be factorised. If `a`, `b`, and `c` are small integers, factorising is usually much faster.
- ✓If the coefficients `a`, `b`, and `c` share a common factor, divide the whole equation by it to simplify the arithmetic. Solving `2x2 - 8x + 6 = 0` is easier as `x2 - 4x + 3 = 0`.
- ✓When calculating the discriminant `b2 - 4ac`, simplify the `4ac` part first. This can prevent arithmetic errors with large numbers under the square root.