# Quadratic Equation Formulas

Solving Equations is the main theme of Algebra. If you need to solve equations of second degree i.e. Quadratic Equations you must be looking for assistance. Take the help of Quadratic Equations Formulas and solve your problems easily and at a faster pace. Learn about the Important Formulae that you might need while solving your math problems.

## List of Quadratic Equation Formulae

If you are looking for some help on how to solve the Quadratic Equations simply check out the list of formulae provided regarding Quadratic Equations. The Quadratic Formulas provided makes it easy for you to calculate the problems corelating to Quadratic Equations. Instead of going with traditional methods to solve Quadratic Equations you can use the simple formulas prevailing and overcome the hassle of doing lengthy calculations.

1. Quadratic Expression

A polynomial of degree two of the form ax^{2} + bx + c (a ≠ 0) is called a quadratic expression in x.

2. The Quadratic Equation

ax^{2} + bx + c = 0 (a ≠ 0) has two roots, given by

α = \(\frac{-b+\sqrt{b^{2}-4 a c}}{2 a}\) and

β = \(\frac{-b-\sqrt{b^{2}-4 a c}}{2 a}\)

3. Nature of roots

The term b^{2} – 4ac is called discriminant of the equation. It is denoted by ∆ or D.

(A) Suppose a, b, c ∈ R and a ≠ 0 then

- If D > 0 ⇒ roots are real and unequal
- If D = 0 ⇒ roots are real and equal and each equal to -b/2a
- If D < 0 ⇒ roots are imaginary and unequal or complex conjugate.

(B) Suppose a, b, c ∈ Q, a ≠ 0 then

- If D > 0 & D is perfect square ⇒ roots are unequal & rational
- If D > 0 & D is not perfect square ⇒ roots are irrational & unequal

4. Conjugate roots

1. If D < 0 →

One root:

α + iβ

Other root:

α – iβ

then

2. D > 0 →

One root:

α + \(\sqrt{\beta}\)

Other root:

α – \(\sqrt{\beta}\)

5. Sum of roots

S = α + β

S = \(\frac{-b}{a}=-\frac{\text { Coefficient of } x}{\text { cofficient of } x^{2}}\)

6. Product of roots

P = αβ

P = \(\frac{c}{a}=\frac{\text { Constant term }}{\text { coefficient of } x^{2}}\)

7. Formation of an equation with given roots

x^{2} – Sx + P = 0

8. Relation between roots-and coefficients

If roots of quadratic equation ax^{2} + bx + c = 0 (a ≠ 0) are α and β then

- (α – β) = \(\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}\) = ± \(\frac{\sqrt{b^{2}-4 a c}}{a}=\frac{\pm \sqrt{D}}{a}\)
- α
^{2}+ β^{2}= (α + β)^{2}– 2αβ = \(\frac{b^{2}-2 a c}{a^{2}}\) - α
^{2}– β^{2}= (α + β)\(\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}\) = – \(\frac{b \sqrt{b^{2}-4 a c}}{a^{2}}=\frac{\pm \sqrt{D}}{a}\) - α
^{3}+ β^{3}= (α + β)^{3}– 3αβ(α + β) = – \(\frac{b\left(b^{2}-3 a c\right)}{a^{3}}\) - α
^{3}– β^{3}= (α – β)^{3}– 3αβ(α – β)\(\sqrt{(\alpha+\beta)^{2}-4 \alpha \beta}\) {(α + β)^{2}– αβ} = \(\frac{\left(b^{2}-a c\right) \sqrt{b^{2}-4 a c}}{a^{3}}\) - α
^{4}+ β^{4}= {(α + β)^{2}– 2αβ}^{2}– 2α^{2}β^{2}= \(\left(\frac{b^{2}-2 a c}{a^{2}}\right)^{2}-2 \frac{c^{2}}{a^{2}}\) - α
^{4}– β^{4}= (α^{2}– β^{2})(α^{2}+ β^{2}) \(=\frac{\pm b\left(b^{2}-2 a c\right) \sqrt{b^{2}-4 a c}}{a^{4}}\) - α
^{2}+ αβ + β^{2}= (α + β)^{2}– αβ - \(\frac{\alpha}{\beta}+\frac{\beta}{\alpha}=\frac{\alpha^{2}+\beta^{2}}{\alpha \beta}=\frac{(\alpha+\beta)^{2}-2 \alpha \beta}{\alpha \beta}\)
- α
^{2}β + β^{2}α = αβ(α + β) - \(\left(\frac{\alpha}{\beta}\right)^{2}+\left(\frac{\beta}{\alpha}\right)^{2}=\frac{\alpha^{4}+\beta^{4}}{\alpha^{2} \beta^{2}}=\frac{\left(\alpha^{2}+\beta^{2}\right)^{2}-2 \alpha^{2} \beta^{2}}{\alpha^{2} \beta^{2}}\)
- nb
^{2}= ac(1 + n)^{2}when one root is n times of another

9. Roots under particular cases

For the quadratic equation ax^{2} + bx + c = 0

- If b = 0 ⇒ roots are of equal magnitude but of opposite sign
- If c = 0 ⇒ one root is zero other is – b/a
- If b = c = 0 ⇒ both roots are zero
- If a = c ⇒ roots are reciprocal to each other
- If \(\left.\begin{array}{ll}

a>0 & c<0 \\

a<0 & c>0

\end{array}\right\}\) ⇒ both roots are of opposite signs - If \(\left.\begin{array}{l}

\mathrm{a}>0, \mathrm{b}>0, \mathrm{c}>0 \\

\mathrm{a}<0, \mathrm{b}<0, \mathrm{c}<0 \end{array}\right\}\) ⇒ both roots are negative - If \(\left.\begin{array}{l} \mathrm{a}>0, \mathrm{b}<0, \mathrm{c}>0 \\

\mathrm{a}<0, \mathrm{b}>0, \mathrm{c}<0

\end{array}\right\}\) ⇒ both roots are positive

10. Condition for common roots

Let quadratic equations are a_{1}x^{2} + b_{1}x + c_{1} = 0 and a_{2}x^{2} + b_{2}x + c_{2} = 0

(i) If only one root is common:

\(\frac{\alpha^{2}}{\mathrm{b}_{1} \mathrm{c}_{2}-\mathrm{b}_{2} \mathrm{c}_{1}}=\frac{\alpha}{\mathrm{a}_{2} \mathrm{c}_{1}-\mathrm{a}_{1} \mathrm{c}_{2}}=\frac{1}{\mathrm{a}_{1} \mathrm{b}_{2}-\mathrm{a}_{2} \mathrm{b}_{1}}\)

(ii) If both roots are common: \(\frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}}\)

11. Nature of the factors of the Quadrate Expression

- Real and different, if b
^{2}– 4ac > 0. - Rational and different, if b
^{2}– 4ac is a perfect square. - Real and equal, if b
^{2}– 4ac = 0.

12. Position for roots of a quadratic equation ax^{2} + bx + c = 0

(A) Condition for both the roots will be greater than k.

(i) D ≥ 0 (ii) k < –\(\frac{b}{2 a}\) (iii) af(k) > 0

(B) Condition for both the roots will be less than k

(i) D ≥ 0 (ii) k > –\(\frac{b}{2 a}\) (iii) af(k) > 0

(C) Condition for k lie between the roots

(i) D > 0 (ii) af(k) < 0

(D) Condition for exactly one root lie in the interval (k_{1}, k_{2}) where k_{1} < k_{2}

(i) f(k_{1}) f(k_{2}) < 0 (ii) D > 0

(E) When both roots lie in the interval (k_{1}, k_{2}) where k_{1} < k_{2}

(i) D > 0 (ii) f(k_{1}) . f(k_{2}) > 0

(F) Any algebraic expression f(x) = 0 in interval [a, b] if

(i) sign of f(a) and f(b) are of same then either no roots or even no. of roots exist.

(ii) sing of f(a) and f(b) are opposite then f(x) = 0 has at least one real root or odd no. of roots.

13. Maximum & Minimum value of Quadratic Expression

In a quadratic expression ax^{2} + bx + c

- If a > 0, quadratic expression has least value at x = –\(\frac{b}{2 a}\). This least value is given by \(\frac{4 a c-b^{2}}{4 a}=-\frac{D}{4 a}\)
- If a < 0, quadratic expression has greatest value at x = –\(\frac{b}{2 a}\). This

greatest value is given by \(\frac{4 a c-b^{2}}{4 a}=-\frac{D}{4 a}\)

14. Quadratic expression in two variables

The general form of a quadratic expression in two variables x & y is ax^{2} + 2hxy + by^{2} + 2gx + 2fy + c. The condition that this expression may be resolved into two linear rational factors is

∆ = \(\left|\begin{array}{lll}

a & h & g \\

h & b & f \\

g & f & c

\end{array}\right|\) = 0 ⇒ abc + 2 fgh – af^{2} – bg^{2} – ch^{2} = 0 and h^{2} – ab > 0

This expression is called discriminant of the above quadratic expression.

### FAQs on Quadratic Equation Formulas

**1. How do you calculate Quadratic Equations easily?**

You can solve the Quadratic Equations easily taking the help of Quadratic Equation Formulas prevailing and save your time.

**2. Where do I get Formulas for Quadratic Equations all at one place?**

You can find Formulas for Quadratic Equations all at one place on our page.