Solving Quadratic Equations
Algebraic techniques for solving quadratic equations, which model parabolic relationships common in physics and engineering, are 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.