In a quadratic equation of the general form \(ax^2 + bx + c = 0\), the roots can be found using the quadratic formula \(x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). The expression under the square root sign, \(b^2 - 4ac\), is called the discriminant, denoted by \(D\). The value of this discriminant determines the nature or types of roots of the quadratic equation without needing to solve for the actual roots first. Furthermore, discriminant conditions allow us to find unknown constants, establish ranges of values for variables, or determine the geometric intersection between straight lines and quadratic curves.
Link the algebraic condition of \(b^2 - 4ac\) to the type of roots and graphical behaviour.
Use the discriminant to determine the nature of roots without solving the full quadratic equation.
For \(ax^2 + bx + c = 0\), where \(a \neq 0\), the discriminant is \(D = b^2 - 4ac\).
When \(b^2 - 4ac > 0\), the graph intersects the \(x\)-axis at two distinct points.
When \(b^2 - 4ac = 0\), the graph touches the \(x\)-axis at exactly one point; the \(x\)-axis is tangent to the curve.
When \(b^2 - 4ac < 0\), the graph does not intersect or touch the \(x\)-axis at all.
When \(b^2 - 4ac \ge 0\), the equation has real roots, encompassing both distinct and equal roots.
When \(b^2 - 4ac < 0\), the square root of a negative number is not real. It is represented by the imaginary unit \(i = \sqrt{-1}\).
Discriminant of a quadratic equation
The value of \(D\) determines the nature of the roots.
Rearrange the given equation into general form \(ax^2 + bx + c = 0\).
Identify coefficients \(a\), \(b\), and constant term \(c\).
Set up the appropriate condition: \(b^2 - 4ac = 0\), \(> 0\), or \(< 0\).
Solve the resulting equation or quadratic inequality.
For \(9x^2 - 6x + 2 = 0\), \(b^2 - 4ac = (-6)^2 - 4(9)(2) = 36 - 72 = -36\).\(x = \dfrac{6 \pm \sqrt{-36}}{18} = \dfrac{6 \pm 6i}{18} = \dfrac{1 \pm i}{3}\).
If \(x^2 - 4ax + 5b = 0\) has two equal real roots, then \((-4a)^2 - 4(1)(5b) = 0\), giving \(16a^2 = 20b\), so \(a^2 = \dfrac{5}{4}b\).
Answer in your mind, then press "Check Answer".
Determine the type of roots for the quadratic equation \(2x^2 - 4x + 5 = 0\).
The quadratic equation \(x^2 + k + 3 = kx\), where \(k\) is a constant, has two equal real roots. Find the possible values of \(k\).
Find the range of values of \(m\) if the quadratic equation \(x^2 + 4x + 13 = m(2 - x)\) has no real roots.
"No real roots" means the quadratic equation has absolutely no mathematical solutions.
It means there are no real number solutions because the square root of a negative number produces imaginary roots involving \(i = \sqrt{-1}\).
A quadratic equation with two equal real roots has only one root.
A quadratic equation always has two roots; when \(b^2 - 4ac = 0\), those two roots are identical (repeated/coincident real roots).
The discriminant \(D\) includes the square root symbol, i.e., \(D = \sqrt{b^2 - 4ac}\).
The discriminant is solely the radicand expression \(b^2 - 4ac\), without the square root sign.
Objective: Identify discriminant conditions, root types and geometric intersections.
1 What is the type of roots for the quadratic equation \(-4x^2 + 4x - 1 = 0\)?
2 If the line \(y = mx - 5\) touches the curve \(2y = x^2 - 1\) at one point, which discriminant condition must be satisfied?
3 When the discriminant \(b^2 - 4ac\) is negative, the square root of a negative number produces an imaginary root represented by \(i = \) ________.
4 For a quadratic equation to have real roots (whether distinct or equal), the required discriminant condition is \(b^2 - 4ac\) ________ \(0\).
Match each discriminant condition or geometric description to its corresponding type of roots or property. On a phone, tap a card first, then tap the matching box.
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