Quadratic Equations: Roots and Discriminant | NIMCET

Quick answer: The roots of $ax^2 + bx + c = 0$ ($a \ne 0$) are $x = \frac{-b \pm \sqrt{D}}{2a}$, where $D = b^2 - 4ac$ is the discriminant. If $D > 0$ the roots are real and distinct, if $D = 0$ they are real and equal, and if $D < 0$ they are complex conjugates. For roots α and β, $\alpha + \beta = -\frac{b}{a}$ and $\alpha\beta = \frac{c}{a}$.

The quadratic formula

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

It comes from completing the square: $a\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a}$.

Nature of roots

Relations between roots and coefficients

Sign of the roots

Common roots

For $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$:

• One common root: $(c_1a_2 - c_2a_1)^2 = (a_1b_2 - a_2b_1)(b_1c_2 - b_2c_1)$.
• Both roots common: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.

Maximum and minimum of a quadratic

For $f(x) = ax^2 + bx + c$, the vertex is at $x = -\frac{b}{2a}$, and the extreme value is

$f\left(-\frac{b}{2a}\right) = -\frac{D}{4a}$

Sign of the expression for all x: $ax^2 + bx + c > 0$ for every real x if and only if $a > 0$ and $D < 0$.

Worked examples

Example 1: nature of roots

Find the nature of the roots of $2x^2 - 5x + 3 = 0$.

$D = 25 - 24 = 1$, a perfect square, so the roots are real, distinct and rational: $x = 1$ and $x = \frac{3}{2}$.

Example 2: equal roots

Find k so that $x^2 + kx + 9 = 0$ has equal roots.

$k^2 - 36 = 0$, so $k = \pm 6$.

Example 3: forming an equation

Form the quadratic with roots $2 + \sqrt{3}$ and $2 - \sqrt{3}$.

Sum $= 4$, product $= 4 - 3 = 1$, so $x^2 - 4x + 1 = 0$.

Example 4: minimum value

Find the minimum value of $x^2 - 6x + 13$.

$D = 36 - 52 = -16$, so the minimum is $-\frac{-16}{4} = 4$, at $x = 3$.

Example 5: positive for all x

For what k is $x^2 + kx + 4 > 0$ for all real x?

Need $D < 0$: $k^2 - 16 < 0$, so $-4 < k < 4$.

Summary table

Exam tips

• Many questions only need $\alpha + \beta$ and $\alpha\beta$; avoid finding the roots.
• Irrational and complex roots of equations with rational (or real) coefficients come in conjugate pairs.
• If $a + b + c = 0$, one root is 1 and the other is $\frac{c}{a}$.

Next, study symmetric functions and location of roots, or see the NIMCET Algebra chapter.

Common mistakes

• Writing $\alpha + \beta = \frac{b}{a}$ without the minus sign.
• Using $D \ge 0$ when the question asks for distinct roots ($D > 0$).
• Forgetting the condition $a \ne 0$ when the leading coefficient contains a parameter.
• Concluding "always positive" from $D < 0$ alone without checking $a > 0$.

Frequently Asked Questions

What is the discriminant of a quadratic equation?

For ax² + bx + c = 0, the discriminant is D = b² − 4ac.

What are the sum and product of the roots?

Sum = −b/a and product = c/a.

When does a quadratic have equal roots?

When the discriminant is zero.

How do you form a quadratic from its roots?

Use x² − (sum of roots)x + (product of roots) = 0.

What is the minimum value of ax² + bx + c when a > 0?

It is −D/(4a), reached at x = −b/(2a).