Discriminant (Δ = b² − 4ac)
Algebra
Intro: Compute Δ and see whether roots are two real, one real (double), or complex.
Worked example
- For $2x^2 - 5x - 3$, find $\Delta$ and interpret.
- Identify coefficients by matching $ax^2 + bx + c$: $$a=2,\; b=-5,\; c=-3.$$
- Discriminant formula: $$\Delta = b^2 - 4ac.$$
- Substitute $a,b,c$: $$\Delta = (-5)^2 - 4(2)(-3).$$
- Compute terms: $$(-5)^2 = 25,\quad -4\cdot 2\cdot (-3) = +24.$$
- Add them: $$\Delta = 25 + 24 = 49.$$
- Interpretation rule: $$\Delta>0 \;\Rightarrow\; \text{two distinct real roots}.$$ Since $49>0$, there are **two real and unequal roots**.
- Optional (roots via quadratic formula): $$x=\frac{-b\pm\sqrt{\Delta}}{2a}=\frac{-(-5)\pm\sqrt{49}}{2\cdot 2}=\frac{5\pm 7}{4}.$$
- Evaluate the two roots: $$x_1=\frac{5+7}{4}=\frac{12}{4}=3,\qquad x_2=\frac{5-7}{4}=\frac{-2}{4}=-\frac{1}{2}.$$
- Optional factorization check: $$2x^2-5x-3= (2x+1)(x-3).$$ Setting each factor to zero gives $x=-\tfrac{1}{2}$ or $x=3$, consistent with $\Delta=49$.
- Conclusion: $$\boxed{\Delta=49}\quad\text{and}\quad\boxed{\text{two distinct real roots}}\;\big(x=3,\;x=-\tfrac{1}{2}\big)$$.
FAQs
What if a=0?
Then it is linear, not quadratic; the discriminant does not apply.
How do signs of Δ classify roots?
Δ>0: two distinct real roots; Δ=0: one real double root; Δ<0: two complex conjugate roots.
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Discriminant
- Match ax^2+bx+c to read off a, b, c
- Δ = b^2 − 4ac
- Δ>0 → two reals; Δ=0 → one double; Δ<0 → complex