2.2.2 Discriminants

Discriminants in Quadratic Equations

The discriminant is a key concept in the study of quadratic equations, typically of the form:

$ax^2 + bx + c = 0$

where $a$, (b), and (c) are constants, and $a \neq 0$. The discriminant is a part of the quadratic formula, which is used to find the roots (solutions) of the quadratic equation:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The discriminant, denoted by the symbol ( \Delta ), is the expression under the square root:

$$\Delta = b^2 - 4ac$$

Interpretation of the Discriminant

The discriminant tells us about the nature of the roots of the quadratic equation:

$$If \Delta > 0 :$$

• The quadratic equation has two distinct real roots.
• The roots are unequal and real.

$$If \Delta = 0 :$$

• The quadratic equation has one real root (or two equal real roots).
• The root is real and repeated.

$$If \Delta < 0 :$$

• The quadratic equation has no real roots.
• The roots are complex (conjugate pairs).

Examples:

Past Edexcel Exam Question:

Solution Outline:

1. Discriminant Condition:

• For two distinct real roots, $\Delta > 0$ .
• The discriminant $\Delta = (2k)^2 - 4(3)(k) = 4k^2 - 12k$.

2. Set Up the Inequality:

• $4k^2 - 12k > 0 .$
• Factorize: $4k(k - 3) > 0 .$

3. Solve the Inequality:

• The critical points are $( k = 0 )$ and $( k = 3 )$.
• The quadratic inequality$\ k(k - 3) > 0$ gives $( k < 0 )$ or $( k > 3 )$.

4. Conclusion:

• The equation has two distinct real roots for$\ ( k < 0 )$ or $( k > 3 )$. This example uses the discriminant to determine the conditions under which a quadratic equation has two distinct real roots.

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Curve Sketching Using the Discriminant

When sketching the graph of a quadratic function $\ y = ax^2 + bx + c ,$the discriminant $\Delta = b^2 - 4ac$ plays a crucial role in determining the nature and number of roots (x-intercepts) of the curve, which is a parabola.

Key Features of the Parabola:

1. Direction:

• $a > 0$: The parabola opens upwards.
• $a < 0$: The parabola opens downwards.

2. Vertex:

• The x-coordinate of the vertex is $\ x = -\frac{b}{2a}$.
• Substitute this x-value into the equation to find the y-coordinate.

3. Y-Intercept:

• The y-intercept is $\ c$, since$\ y = c$ when $x = 0 .$

4. X-Intercepts:

• The discriminant $\Delta = b^2 - 4ac$ determines the number of x-intercepts:
• $\Delta > 0$: Two distinct real roots (two x-intercepts).
• $\Delta = 0$: One real root (the vertex touches the x-axis).
• $\Delta < 0$: No real roots (the parabola does not touch the x-axis).

Steps for Sketching the Curve:

1. Determine the Direction:

• Check the sign of $\ a$ to see if the parabola opens upwards or downwards.

2. Find the Vertex:

• Calculate the vertex using $x = -\frac{b}{2a}$ and find the corresponding y-coordinate.

3. Plot the Y-Intercept:

• The y-intercept is at $\ (0, c) .$

4. Analyse the Discriminant:

• Calculate $\Delta = b^2 - 4ac$ to determine the number of x-intercepts.
• If $\ \Delta > 0$, solve for the roots using the quadratic formula to find the x-intercepts.

5. Plot Key Points:

• Plot the vertex, $y-intercept$, and any $x-intercepts$.
• Draw a smooth curve through these points, ensuring it opens in the correct direction.

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