Discriminant

Japanese school year: Math I

What you learn

You will learn how to determine the number of real roots without solving the equation by using the discriminant. It is widely used to analyze intersections between graphs and lines or to find parameter conditions in inequalities. Understanding the structure of the quadratic formula beforehand is helpful.

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Key points

This formula (the discriminant) determines how many real solutions an equation has without solving it. Check whether $D$ is positive, zero, or negative.

D = b^2 - 4ac

Use this simpler formula when the coefficient of $x$ is even. Substituting half of that coefficient as $b'$ keeps calculations smaller.

\frac{D}{4} = b'^2 - ac

This formula directly finds the solution $x$ when there is only one real root (a double root). When $D = 0$ , the square root part disappears.

D = 0 \implies x = -\frac{b}{2a}

This rule tells you how many real solutions exist based on $D$ . There are two solutions if $D > 0$ , one (a double root) if $D = 0$ , and no real solutions if $D < 0$ .

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Practice 1Coming soon