More about equations
How to solve one linear and one quadratic equation by graph and by algebra, and fractional, exponential, logarithmic and trigonometric equations that become quadratics. The unit is NF.
What you will learn
- 5.1 Solve one linear and one quadratic equation (y = ax² + bx + c) by graph.
- 5.2 Solve them algebraically.
- 5.3 Solve equations that become quadratics: fractional, exponential, logarithmic and trigonometric (0°–360°).
- 5.4 Word problems with these equations.
You need first: Quadratic equations in one unknown, Exponential and logarithmic functions, Trigonometry (2-D)
Key terms
| English | 中文 |
|---|---|
| simultaneous equations (one linear, one quadratic) | 聯立方程（一條一次，一條二次） |
| fractional equation | 分式方程 |
| exponential / logarithmic equation | 指數方程／對數方程 |
| trigonometric equation | 三角方程 |
| substitution (let u = …) | 代換（設 u = ……） |
| reject | 捨去 |
Standard steps
One linear, one quadratic
- Make y (or x) the subject of the linear equation.
- Substitute into the quadratic.
- Solve the quadratic.
- Put each x back into the linear equation to get y. Pair them up.
"Hidden quadratic": let u = …
- Spot the repeated block: x² in x⁴ − 5x² + 4; 2ˣ in 4ˣ − 5·2ˣ + 4 (because 4ˣ = (2ˣ)²); sin θ in 2 sin²θ − sin θ − 1.
- Let u = that block. Rewrite as a quadratic in u.
- Solve for u.
- Go back: solve "block = each value of u".
- Reject values that are impossible: 2ˣ > 0, −1 ≤ sin θ ≤ 1, inside of a log > 0, denominator ≠ 0.
Fractional equations
- Multiply both sides by the L.C.M. of the denominators.
- Solve the quadratic.
- Reject any answer that makes a denominator 0.
Worked examples
Example 1: Solve y = x + 1 and x² + y² = 25.
x² + (x + 1)² = 25
2x² + 2x − 24 = 0
x² + x − 12 = 0
(x + 4)(x − 3) = 0
x = 3, y = 4 or x = −4, y = −3Example 2: Solve 4ˣ − 5(2ˣ) + 4 = 0.
let u = 2ˣ: u² − 5u + 4 = 0
(u − 1)(u − 4) = 0
2ˣ = 1 → x = 0; 2ˣ = 4 → x = 2Example 3: Solve log x + log(x − 3) = 1.
log[x(x − 3)] = 1
x² − 3x = 10
(x − 5)(x + 2) = 0
x = 5 (reject x = −2: log of a negative number is undefined)Example 4: Solve 1/x + 1/(x + 2) = 3/4.
× 4x(x + 2): 4(x + 2) + 4x = 3x(x + 2)
3x² − 2x − 8 = 0
(3x + 4)(x − 2) = 0
x = 2 or x = −4/3Example 5: Solve 2 sin²θ − sin θ − 1 = 0 for 0° ≤ θ ≤ 360°.
(2 sin θ + 1)(sin θ − 1) = 0
sin θ = 1 → θ = 90°
sin θ = −1/2 → θ = 210° or 330°Common mistakes
- Substituting x back into the quadratic and getting extra, wrong y-values.
- Forgetting to go back from u to x.
- Keeping an impossible value, such as 2ˣ = −1 or sin θ = 2.
- Not checking log answers in the original equation.
- Multiplying only some terms by the L.C.M.
How the DSE tests it
- Section B: a log or exponential equation reduced to a quadratic, often as part (b) after part (a) sets up a relation.
- Section B: a line meeting a parabola or a circle.
- Paper 2: hidden quadratics in exponential or trigonometric form.
Questions you might not ask in class
How do I spot the hidden quadratic?
Look for something and its square: x and x², 2ˣ and 4ˣ, sin θ and sin²θ.
Why do we reject answers? Didn't I solve it right?
You did. But changing the equation (multiplying, removing logs) can add answers that don't fit the original. Checking removes them.
Why substitute into the linear equation, not the quadratic?
The linear one gives one y for each x. The quadratic can give two, and one would be wrong.