Class 10 Maths Quadratic Equations Notes

4.1 Introduction to Quadratic Equations

A quadratic equation is a second-degree polynomial equation in one variable, which can be expressed in the general form:$$ax^2 + bx + c = 0$$ax2+bx+c=0

Where:

• $a, b, c$a,b,c are constants (with $a \neq 0$a=0),
• $x$x is the variable.

The degree of the equation is 2 because the highest power of $x$x is $x^2$x2. Quadratic equations are important in many areas of mathematics and real-world applications, such as projectile motion, profit and loss calculations, and engineering.

Examples of quadratic equations include:

1. $x^2 – 5x + 6 = 0$x2−5x+6=0
2. $2x^2 + 3x – 5 = 0$2×2+3x−5=0

The solution to a quadratic equation is called the root of the equation.

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4.2 Quadratic Equations: Types and Forms

A quadratic equation is generally written as:$$ax^2 + bx + c = 0$$ax2+bx+c=0

Depending on the values of $a$a, $b$b, and $c$c, quadratic equations can be classified into different types:

1. Standard Form: $ax^2 + bx + c = 0$ax2+bx+c=0
2. Factored Form: $a(x – p)(x – q) = 0$a(x−p)(x−q)=0, where $p$p and $q$q are the roots of the equation.
3. Vertex Form: $a(x – h)^2 + k = 0$a(x−h)2+k=0, where $(h, k)$(h,k) is the vertex of the parabola.

Nature of the Roots:

• Real and Equal Roots: When the discriminant $\Delta = b^2 – 4ac = 0$Δ=b2−4ac=0.
• Real and Distinct Roots: When the discriminant $\Delta > 0$Δ>0.
• Complex Roots: When the discriminant $\Delta < 0$Δ<0.

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4.3 Solution of a Quadratic Equation by Factorisation

One of the simplest methods to solve a quadratic equation is by factorisation. The steps are as follows:

1. Write the quadratic equation in standard form: $ax^2 + bx + c = 0$ax2+bx+c=0.
2. Find two numbers that multiply to give $a \times c$a×c (product) and add up to give $b$b (sum).
3. Split the middle term $bx$bx into two terms using the two numbers found in step 2.
4. Factor by grouping: Group the terms in pairs and factor out the common terms.
5. Solve for xxx by setting each factor equal to zero.

Example 1:

Solve $x^2 – 5x + 6 = 0$x2−5x+6=0 by factorisation.

1. We need two numbers that multiply to $1 \times 6 = 6$1×6=6 and add to $-5$−5. These numbers are $-2$−2 and $-3$−3.
2. Split the middle term: $$x^2 – 2x – 3x + 6 = 0$$x2−2x−3x+6=0
3. Group terms: $$x(x – 2) – 3(x – 2) = 0$$x(x−2)−3(x−2)=0
4. Factor out the common factor: $$(x – 2)(x – 3) = 0$$(x−2)(x−3)=0
5. Set each factor equal to zero: $$x – 2 = 0 \quad \text{or} \quad x – 3 = 0$$x−2=0orx−3=0 So, $x = 2$x=2 or $x = 3$x=3.

Thus, the roots of the equation are $x = 2$x=2 and $x = 3$x=3.

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4.4 Nature of Roots

The nature of the roots of a quadratic equation is determined by the discriminant, denoted by $\Delta$Δ, which is given by:$$\Delta = b^2 – 4ac$$Δ=b2−4ac

The discriminant helps determine the type of roots for a given quadratic equation:

• If Δ>0\Delta > 0Δ>0: The quadratic equation has two distinct real roots.
• If Δ=0\Delta = 0Δ=0: The quadratic equation has one real root (also called equal roots).
• If Δ<0\Delta < 0Δ<0: The quadratic equation has no real roots (the roots are complex).

Example 2:

Consider the quadratic equation $2x^2 + 3x – 5 = 0$2×2+3x−5=0.

• Here, $a = 2$a=2, $b = 3$b=3, and $c = -5$c=−5.
• The discriminant is: $$\Delta = b^2 – 4ac = 3^2 – 4(2)(-5) = 9 + 40 = 49$$Δ=b2−4ac=32−4(2)(−5)=9+40=49
• Since $\Delta > 0$Δ>0, the equation has two distinct real roots.

Example 3:

For the equation $x^2 + 4x + 4 = 0$x2+4x+4=0:

• Here, $a = 1$a=1, $b = 4$b=4, and $c = 4$c=4.
• The discriminant is: $$\Delta = b^2 – 4ac = 4^2 – 4(1)(4) = 16 – 16 = 0$$Δ=b2−4ac=42−4(1)(4)=16−16=0
• Since $\Delta = 0$Δ=0, the equation has one real root (repeated root).

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4.5 Summary

In this chapter, we have covered the key concepts of quadratic equations, including their general form, the methods to solve them, and the nature of their roots:

• A quadratic equation is an equation of degree 2, of the form $ax^2 + bx + c = 0$ax2+bx+c=0.
• The roots of a quadratic equation can be found using factorisation, quadratic formula, or completing the square (not covered in detail here).
• The discriminant$\Delta = b^2 – 4ac$Δ=b2−4ac helps determine the nature of the roots:
  • Real and distinct roots if $\Delta > 0$Δ>0,
  • Real and equal roots if $\Delta = 0$Δ=0,
  • No real roots if $\Delta < 0$Δ<0.

By mastering these concepts, you will be well-equipped to solve quadratic equations and analyze their roots efficiently.

MCQs Based on the “Quadratic Equations” Chapter:

1. Which of the following is the standard form of a quadratic equation?

a) $ax^2 + bx + c = 0$ax2+bx+c=0
b) $ax^2 + b = 0$ax2+b=0
c) $ax + b = 0$ax+b=0
d) $a^2x + b = 0$a2x+b=0

Answer: a) $ax^2 + bx + c = 0$ax2+bx+c=0

2. In the quadratic equation x2−5x+6=0x^2 – 5x + 6 = 0x2−5x+6=0, the roots are:

a) $1$1 and $6$6
b) $2$2 and $3$3
c) $3$3 and $2$2
d) $5$5 and $-6$−6

Answer: b) $2$2 and $3$3

3. What is the discriminant of the quadratic equation 3×2−4x+2=03x^2 – 4x + 2 = 03×2−4x+2=0?

a) $16 – 24 = -8$16−24=−8
b) $16 – 24 = 8$16−24=8
c) $9 – 24 = -15$9−24=−15
d) $16 – 18 = -2$16−18=−2

Answer: a) $16 – 24 = -8$16−24=−8

4. Which of the following statements is true for the quadratic equation x2+4x+4=0x^2 + 4x + 4 = 0x2+4x+4=0?

a) It has two real roots.
b) It has one real root.
c) It has no real roots.
d) It has complex roots.

Answer: b