Unit 5
Algebra
Class 12 Mathematics
Quadratic Equations
Class 12 Mathematics – Quadratic Equations Notes PDF
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NEB / CDC Focus
This chapter focuses on the nature and roots of a quadratic equation, the relation between roots and coefficients, formation of a quadratic equation from given roots, symmetric functions of the roots, and conditions involving one or both roots common to two quadratic equations.
1. Quadratic Equation
An equation of the form ax²+bx+c=0, where a,b,c are constants and a≠0, is called a quadratic equation.
A value of x that satisfies the equation is called a root or solution of the quadratic equation. A quadratic equation has two roots when roots are counted with multiplicity and complex roots are allowed.
| Expression | Quadratic? | Reason |
|---|---|---|
| 2x²−5x+3=0 | Yes | Highest power of x is 2 and the coefficient of x² is non-zero. |
| x²−9=0 | Yes | The coefficient of x may be zero. |
| 3x+7=0 | No | Degree is 1. |
| x³+x−1=0 | No | Degree is 3. |
2. Finding the Roots of a Quadratic Equation
2.1 Factorization
When possible, express the quadratic as a product of two linear factors and apply the zero-product rule.
Solve x²−5x+6=0.
x²−5x+6=(x−2)(x−3)=0.
Therefore x=2 or x=3.
2.2 Completing the Square and the Quadratic Formula
Start with
ax²+bx+c=0, a≠0.
- Divide by a: x²+(b/a)x+c/a=0.
- Move the constant: x²+(b/a)x=−c/a.
- Add (b/2a)² to both sides.
- Obtain (x+b/2a)²=(b²−4ac)/(4a²).
- Take square roots and isolate x.
3. Nature of the Roots
For ax²+bx+c=0, the quantity D=b²−4ac is called the discriminant.
| Discriminant | Nature of roots | Graphical meaning |
|---|---|---|
| D>0 | Two distinct real roots | Parabola cuts the x-axis at two points. |
| D=0 | Two equal real roots | Parabola touches the x-axis at one point. |
| D<0 | No real roots; two non-real complex conjugate roots for real coefficients | Parabola does not meet the x-axis. |
When a,b,c are rational numbers and D>0, the roots are rational if D is a perfect square of a rational number; otherwise the real roots are generally irrational.
For 2x²−4x+5=0,
D=(−4)²−4(2)(5)=16−40=−24<0.
Therefore the equation has two non-real complex conjugate roots.
4. Relation Between Roots and Coefficients
Let α and β be the roots of ax²+bx+c=0. Then
ax²+bx+c = a(x−α)(x−β).
Expanding the right side:
a[x²−(α+β)x+αβ].
Comparing coefficients gives:
4.1 Useful Consequences
| Expression | In terms of S=α+β and P=αβ |
|---|---|
| α²+β² | S²−2P |
| α³+β³ | S³−3PS |
| (α−β)² | S²−4P |
| 1/α+1/β | S/P, provided P≠0 |
| 1/α²+1/β² | (S²−2P)/P², provided P≠0 |
| α/β+β/α | (S²−2P)/P, provided P≠0 |
5. Formation of a Quadratic Equation
If α and β are the required roots, then
Any non-zero constant multiple represents the same roots: k[x²−(α+β)x+αβ]=0, k≠0.
Form the quadratic equation whose roots are 3+√2 and 3−√2.
S=(3+√2)+(3−√2)=6,
P=(3+√2)(3−√2)=9−2=7.
Hence the equation is
x²−6x+7=0.
6. Symmetric Functions of the Roots
An expression in α and β is symmetric if it remains unchanged when α and β are interchanged.
Examples include α+β, αβ, α²+β² and α³+β³. Such expressions can usually be rewritten using S=α+β and P=αβ.
If α,β are roots of 2x²−7x+3=0, find α²+β².
S=α+β=7/2, P=αβ=3/2.
α²+β²=S²−2P=49/4−3=37/4.
7. Formation of Equations with Transformed Roots
If α,β are roots of a known quadratic, transformed-root questions can often be solved by finding the new sum and product rather than solving for α and β separately.
| New roots | New sum | New product |
|---|---|---|
| α+k, β+k | S+2k | P+kS+k² |
| kα, kβ | kS | k²P |
| 1/α, 1/β | S/P | 1/P |
| α², β² | S²−2P | P² |
| −α, −β | −S | P |
If α,β are roots of 2x²−5x+3=0, form the equation whose roots are 1/α,1/β.
S=5/2, P=3/2.
New sum=S/P=5/3, new product=1/P=2/3.
x²−(5/3)x+2/3=0.
Multiplying by 3:
3x²−5x+2=0.
8. Quadratic Equations Having One or Both Roots Common
Consider
a₁x²+b₁x+c₁=0
a₂x²+b₂x+c₂=0.
8.1 Both Roots Common
If the two equations have both roots common, they represent the same quadratic up to multiplication by a non-zero constant. Therefore their corresponding coefficients are proportional.
Use cross-multiplication if a denominator is zero.
8.2 One Root Common
If the equations have exactly one common root, eliminate the x² terms to obtain the possible common root:
This form assumes the displayed denominator is non-zero.
A useful coefficient condition for the existence of a common root is
If this condition holds but the three coefficient ratios are not all equal, the two quadratics have one common root rather than both roots common.
Find the common root of x²−5x+6=0 and x²−4x+3=0.
Subtract the second equation from the first:
−x+3=0 ⇒ x=3.
The common root is 3.
Compare 2x²−10x+12=0 and x²−5x+6=0.
2/1 = (−10)/(−5) = 12/6 = 2.
The equations are proportional, so both roots are common.
9. Worked Examples
Solve 3x²−2x−1=0.
x=[2±√{(−2)²−4(3)(−1)}]/6 =[2±√16]/6 =[2±4]/6.
Hence x=1 or x=−1/3.
Find k if x²+kx+9=0 has equal roots.
For equal roots, D=0.
k²−4(1)(9)=0 ⇒ k²=36 ⇒ k=±6.
Form the quadratic equation whose roots are 2 and −5.
S=−3, P=−10.
x²−Sx+P=0 ⇒ x²+3x−10=0.
If α,β are roots of x²−4x+1=0, then S=4, P=1.
α³+β³=S³−3PS=64−12=52.
Let α,β be roots of x²−6x+5=0.
S=6, P=5.
New sum=α²+β²=S²−2P=36−10=26.
New product=α²β²=P²=25.
Required equation:
x²−26x+25=0.
If the roots of ax²+bx+c=0 are equal in magnitude but opposite in sign, then α+β=0.
−b/a=0 ⇒ b=0.
Thus a quadratic with opposite roots has no linear term.
If the roots are reciprocals, then αβ=1.
c/a=1 ⇒ c=a.
Hence the condition is a=c.
Suppose x=2 is a common root of x²−5x+6=0 and x²+kx−2=0. Find k.
Substitute x=2 into the second equation:
4+2k−2=0 ⇒ 2k+2=0 ⇒ k=−1.
10. Problem-Solving Strategy
- Write the equation in standard form ax²+bx+c=0.
- If only the nature of roots is needed, calculate D=b²−4ac; do not solve unnecessarily.
- If expressions in roots are required, first use S=−b/a and P=c/a.
- To form an equation from roots, use x²−Sx+P=0.
- For transformed roots, calculate their new sum and product.
- For common-root problems, eliminate one power of x or use the coefficient condition where useful.
- Check whether “one common root” and “both roots common” are being distinguished.
- Substitute a final numerical root back into the original equation to catch sign errors.
11. Common Mistakes and Warnings
If a=0, the equation is not quadratic.
α+β=−b/a, not b/a.
Use b²−4ac; keep the sign of b inside the square correctly.
D=0 means equal real roots; D<0 means non-real roots for real coefficients.
Symmetric-root questions are usually shorter with S and P.
The product of 1/α and 1/β is 1/P.
Both common roots require proportional quadratic equations, not merely one shared solution.
Keep one coefficient notation consistently throughout elimination.
12. Exam-Important Formula Sheet
13. Important Exam Questions
Short-Answer Questions
- Define a quadratic equation and its roots.
- State the quadratic formula.
- Define the discriminant of ax²+bx+c=0.
- State the nature of roots for D>0, D=0, and D<0.
- If α,β are roots, state α+β and αβ.
- Form the equation whose roots are two given numbers.
- State the condition for both roots of two quadratics to be common.
- Express α²+β² in terms of α+β and αβ.
Derivation / Proof Questions
- Derive the quadratic formula by completing the square.
- Establish the relation between the roots and coefficients of a quadratic equation.
- Derive the equation whose roots are α and β.
- Derive useful symmetric expressions such as α²+β² and α³+β³.
- Derive a condition for two quadratic equations to possess a common root.
Numerical / Application Questions
- Solve a quadratic equation by factorization or the quadratic formula.
- Determine the nature of roots without solving the equation.
- Find a parameter so that a quadratic has equal, distinct or non-real roots.
- Find symmetric functions of roots without calculating the roots individually.
- Form an equation whose roots are shifted, multiplied, squared or reciprocal forms of the original roots.
- Find a common root of two quadratic equations.
- Determine whether two quadratics have one root or both roots common.
Diagram Questions
- Draw parabolas illustrating D>0, D=0, and D<0.
- Use a diagram to explain the relationship between roots and x-axis intersections.
- Draw a flowchart for forming a quadratic equation from its roots.
14. One-Minute Revision
- A quadratic equation has standard form ax²+bx+c=0 with a≠0.
- The quadratic formula is x=[−b±√(b²−4ac)]/(2a).
- The discriminant is D=b²−4ac.
- D>0: two distinct real roots.
- D=0: equal real roots.
- D<0: non-real conjugate roots for real coefficients.
- If roots are α,β, then α+β=−b/a.
- The product is αβ=c/a.
- An equation with roots α,β is x²−(α+β)x+αβ=0.
- Symmetric expressions should usually be rewritten using the sum and product of roots.
- α²+β²=S²−2P.
- α³+β³=S³−3PS.
- For reciprocal roots, new sum is S/P and new product is 1/P.
- If both roots are common, the two equations are proportional.
- For common-root questions, careful elimination is often the fastest method.
15. Diagram Practice
- Parabola showing two real roots as x-axis intersections.
- Quadratic-formula derivation flowchart.
- Three discriminant cases: two, one and no real x-intercepts.
- Factor-form to coefficient-form relation for roots and coefficients.
- Formation-of-equation flowchart using sum and product.
- Symmetric-functions map using S=α+β and P=αβ.
- One-common-root versus both-roots-common diagram.
Also Visit
Original Nepal eNotes page: Class 12 Mathematics Quadratic Equations Notes.
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