A quadratic equation is a polynomial equation where the highest power of the variable is 2. In general form, it is written as \(ax^2 + bx + c = 0\), where \(a, b,\) and \(c\) are constants and \(a \neq 0\). The solutions or roots of a quadratic equation represent the \(x\)-coordinates of the intersection points between the curve \(y = ax^2 + bx + c\) and the \(x\)-axis. Beyond basic factorisation, quadratic equations can be solved using completing the square or the quadratic formula. Furthermore, when roots are given, new quadratic equations can be formed using the sum and product of roots. Finally, quadratic inequalities can be solved using graph sketching, number line, or table methods.
Compare graph sketching, number line and table methods to identify solution intervals.
Sketch an upward-opening parabola with roots \(-1\) and \(3\). For \(> 0\), choose the shaded positive wings: \(x < -1\) or \(x > 3\). For \(< 0\), choose the dip below the \(x\)-axis: \(-1 < x < 3\).
Mark critical values \(-1\) and \(3\). Test points \(-2\), \(0\), and \(4\) give signs \(+\), \(-\), \(+\) across the intervals \(x < -1\), \(-1 < x < 3\), and \(x > 3\).
Quadratic equations can always be written in the form \(ax^2 + bx + c = 0\).
General form: \(ax^2 + bx + c = 0\), where \(a, b, c\) are constants and \(a \neq 0\). The roots are values of \(x\) when \(y = 0\), which correspond to the \(x\)-intercepts of \(y = ax^2 + bx + c\).
If \(a \neq 1\), divide both sides by \(a\) so the coefficient of \(x^2\) is 1. Move the constant term to the right-hand side, add \(\left(\dfrac{\text{coefficient of } x}{2}\right)^2\) to both sides, factorise into \((x + p)^2\), then take the square root of both sides including \(\pm\).
\(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). Brahmagupta produced an equivalent formula: \(x = \dfrac{\sqrt{4ac + b^2} - b}{2a}\). Al-Khwarizmi used methods equivalent to completing the square.
If \(\alpha\) and \(\beta\) are the roots: \((x - \alpha)(x - \beta) = 0 \implies x^2 - (\alpha + \beta)x + \alpha\beta = 0\).
Sum of roots (SOR): \(\alpha + \beta = -\dfrac{b}{a}\). Product of roots (POR): \(\alpha\beta = \dfrac{c}{a}\).
For \((x - a)(x - b) > 0\) with \(a < b\): \(x < a\) or \(x > b\). For \((x - a)(x - b) < 0\): \(a < x < b\).
If \(a \neq 1\), divide both sides by \(a\) so the coefficient of \(x^2\) is 1.
Move the constant term to the right-hand side.
Add \(\left(\dfrac{\text{coefficient of } x}{2}\right)^2\) to both sides.
Factorise the left-hand side into \((x + p)^2\), then take square roots and solve for \(x\).
Quadratic formula
Use when factorisation is not simple.
Equation from roots
Use the sum and product of roots.
Answer in your mind, then press "Check Answer".
Solve the quadratic equation \(x^2 + 4x - 7 = 0\) by completing the square. Give the answers correct to three decimal places.
If \(\alpha\) and \(\beta\) are roots of \(2x^2 + x - 4 = 0\), form a quadratic equation with roots \(2\alpha\) and \(2\beta\).
Find the range of values of \(x\) for the inequality \(x^2 - 3x - 4 \ge 0\).
Many students believe \((x - a)(x - b) > 0\) means \((x - a) > 0\) AND \((x - b) > 0\) only.
The product of two factors is positive if both are positive OR both are negative. Hence, \(x > b\) (both positive) or \(x < a\) (both negative).
Thinking that the roots of an equation change when the entire equation is multiplied by a non-zero constant.
Multiplying both sides of \(ax^2 + bx + c = 0\) by a constant \(k \neq 0\) changes the coefficients, but the roots remain unchanged.
Assuming that completing the square can only be used when the coefficient of \(x^2\) is 1.
It can be used for any quadratic equation, but you must first divide the entire equation by \(a\) so that the coefficient of \(x^2\) becomes 1.
Objective: Test your understanding of roots, formulas and inequality solution rules.
1 What is the sum of roots for the quadratic equation \(3x^2 - 9x + 5 = 0\)?
2 Which of the following is the solution range for the quadratic inequality \((x + 2)(x - 5) < 0\)?
3 In general form, the formula used to calculate the product of roots \(\alpha\beta\) from \(ax^2 + bx + c = 0\) is ________.
4 When solving a quadratic equation by completing the square after ensuring the coefficient of \(x^2\) is 1, the term added to both sides of the equation is the square of half the ________ of \(x\).
Match each mathematical concept or inequality form to its correct formula or solution rule. On a phone, tap a term first, then tap the matching box.
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