The Discriminant and Complex Roots
The expression $${b^2} – 4ac$$, which appears under the radical sign in the quadratic formula
\[x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}\]
is called the discriminant of the quadratic equation $$a{x^2} + bx + c = 0$$.
If $$a$$, $$b$$ and $$c$$ are real numbers, you can use the algebraic sign of the discriminant to determine the number and the nature of the roots of the quadratic equation.
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If $${b^2} – 4ac$$ is positive, the equation has two real and unequal roots.
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If $${b^2} – 4ac = 0$$, the equation has only one root, i.e., double roots.
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If $${b^2} – 4ac$$ is negative, the equation has no real root. Its roots are two complex numbers that are complex conjugates of each other.
Example:
Use the discriminant to determine the nature of the roots of each quadratic equation without actually solving it.
(a) $$5{x^2} – x – 3 = 0$$
(b) $$9{x^2} + 42x + 49 = 0$$
(c) $${x^2} – x + 1 = 0$$
Solution:
(a) Here $$a = 5$$, $$b = – 1$$, $$c = – 3$$ and $${b^2} – 4ac = {\left( { – 1} \right)^2} – 4\left( 5 \right)\left( { – 3} \right) = 61$$ is positive, hence there are two unequal real roots.
(b) Here $$a = 9$$, $$b = 42$$, $$c = 49$$ and $${b^2} – 4ac = {\left( {42} \right)^2} – 4\left( 9 \right)\left( {49} \right) = 0$$, hence the equation has just one root: a double root and this root is a real number.
(c) Here $$a = 1$$, $$b = – 1$$, $$c = 1$$ and $${b^2} – 4ac = {\left( { – 1} \right)^2} – 4\left( 1 \right)\left( 1 \right) = – 3$$ is negative, hence the equation has no real root.
Example:
Use the quadratic formula to find the roots of the quadratic equation $${x^2} – x + 1 = 0$$.
Solution:
Using the quadratic formula, with $$a = 1$$, $$b = – 1$$ and $$c = 1$$, we have
$$x = \frac{{ – b \pm \sqrt {{b^2} – 4ac} }}{{2a}}$$
$$x = \frac{{ – \left( { – 1} \right) \pm \sqrt {{{\left( { – 1} \right)}^2} – 4\left( 1 \right)\left( 1 \right)} }}{{2\left( 1 \right)}}$$
$$ = \frac{{1 \pm \sqrt { – 3} }}{2}$$
$$ = \frac{{1 \pm \sqrt 3 i}}{2} = \frac{1}{2} \pm \frac{{\sqrt 3 }}{2}i$$
Thus, the roots are the complex conjugates $$\frac{1}{2} + \frac{{\sqrt 3 }}{2}i$$ and $$\frac{1}{2} – \frac{{\sqrt 3 }}{2}i$$