Notes: Quadratic Equations | Mathematics (Maths) for JEE Main & Advanced PDF Download

What are quadratic equations?

A quadratic equation is an equation with a variable to the second power as its highest power term. For example, in the quadratic equation 3x² - 5x-2=0:

• x is the variable, which represents a number whose value we don't know yet.
• The 2 is the power or exponent. An exponent of 2 means the variable is multiplied by itself.
• 3 and -5 are the coefficients, or constant multiples of x² and x. 3x² is a single term, as is -5x.
• -2 is a constant term.

How do I solve quadratic equations using square roots?

When can I solve by taking square roots?

• Quadratic equations without x-terms such as 2x² = 32 can be solved without setting a quadratic expression equal to 0. Instead, we can isolate x² and use the square root operation to solve for x.
• When solving quadratic equations by taking square roots, both the positive and negative square roots are solutions to the equation. This is because when we square a solution, the result is always positive.

For example, for the equation x² = 4, both 2 and -2 are solutions:

• 2² = 4
• (-2)² = 4

When solving quadratic equations without x-terms:

• Isolate x².
• Take the square root of both sides of the equation. Both the positive and negative square roots are solutions.

Example: What values of x satisfy the equation 2x² = 18?
Sol: 

The following values of x satisfy the equation 2x² = 18:

• -3 and 3

Zeroes and Roots: When referring to the zeroes of $ax\mathrm{<sup\; style="">2</sup>}+bx+cax^2\; +\; bx\; +\; c$, it usually means the roots of the equation $ax\mathrm{<sup\; style="">2</sup>}+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$.

Nature of Roots

A quadratic equation has exactly two roots, which can be:

• Real and distinct
• Real and equal
• Complex (imaginary)

The Quadratic formula

Not all quadratic expressions are factorable, and not all factorable quadratic expressions are easy to factor. The quadratic formula gives us a way to solve any quadratic equation as long as we can plug the correct values into the formula and evaluate.

What are the steps?
To solve a quadratic equation using the quadratic formula:

• Rewrite the equation in the form ax² + bx + c = 0.
• Substitute the values of a, b, and c into the quadratic formula, shown below.
  
• Evaluate x.

Example:  Solve the quadratic equation 2x² + 4x + 2 = 0.
Solution:  Compare 2x² + 4x + 2 = 0 with ax² + bx + c = 0.

Now, use the Quadratic Formula, 

We have x equals -1

Discriminant

The term $b\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}-4acb^2\; -\; 4ac$ is called the discriminant ($\backslash De$Δ):

• If $\Delta >0\backslash Delta\; >\; 0$, there are two distinct real roots.
• If $\Delta =0\backslash Delta\; =\; 0$, there are two equal real roots.
• If , there are two complex roots.

Roots

For a quadratic equation $ax\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$ with roots $\alpha \backslash alpha$ and $\beta \backslash beta$:

1. The sum of the roots: $\alpha +\beta =-\frac{b}{a}\backslash alpha\; +\; \backslash beta\; =\; -\backslash frac\{b\}\{a\}$
2. The product of the roots: $\alpha \beta =\frac{c}{a}\backslash alpha\; \backslash beta\; =\; \backslash frac\{c\}\{a\}$
3. The difference of the roots: 

Forming Quadratic Equations : Given roots $\alpha \backslash alpha$ and $\beta \backslash beta$, the quadratic equation can be constructed as: 

$(x-\alpha )(x-\beta )=0(x\; -\; \backslash alpha)(x\; -\; \backslash beta)\; =\; 0$ .

This simplifies to: $x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}-(\alpha +\beta )x+\alpha \beta =0x^2\; -\; (\backslash alpha\; +\; \backslash beta)x\; +\; \backslash alpha\; \backslash beta\; =\; 0$  Or in terms of the coefficients: $x\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}-(-\frac{b}{a})x+\frac{c}{a}=0$

Example: If a, b are the roots of a quadratic equation x² - 3x + 5 = 0, then find the equation whose roots are (a² - 3a + 7) and (b² - 3b + 7). 

Solution:  Since a & b are the roots of the equation x² - 3x + 5 = 0, 

So, a² - 3a + 5 = 0.  and b² - 3b + 5 = 0.
Hence, we have a² - 3a = -5 and b² - 3b = -5.
Putting these values in the new roots.
We have new roots as  -5+7 , -5+7
2 and 2 are the roots .

Hence, the required equation is: x² - 4x + 4 = 0.

Nature of the Roots

Consider the quadratic equation $ax\mathrm{<sup\; style="">2</sup>}+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$ where $a,b,c\in Ra,\; b,\; c\; \backslash in\; \backslash mathbb\{R\}$and $a\ne 0$. The roots can be determined using the quadratic formula: The nature of the roots depends on the value of $DD$ (Discriminant):

1. $D>0D\; >\; 0$: The roots are real and distinct (unequal).
2. $D=0D\; =\; 0$: The roots are real and coincident (equal).
3. : The roots are complex (imaginary).
4. If one root is $p+iqp\; +\; iq$, then the other root is its conjugate $p-iqp\; -\; iq$(where $p,q\in Rp,\; q\; \backslash in\; \backslash mathbb\{R\}$ and $i=-1i\; =\; \backslash sqrt\{-1\}$).

Roots in Rational Quadratic Equations

For a quadratic equation $ax\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$ where $a,b,c\in Qa,\; b,\; c\; \backslash in\; \backslash mathbb\{Q\}$ and $a\ne 0a\; \backslash neq\; 0$:

1. If $DD$ is a perfect square, the roots are rational.
2. If one root is $p+qp\; +\; q$ (where $pp$ is rational and $qq$ is a surd), the other root will be $p-qp\; -\; q$.

Properties of Roots in Special Cases

For the quadratic equation $ax\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$with real roots:

1. $b=0b\; =\; 0$: The roots are equal in magnitude but opposite in sign.
2. $c=0c\; =\; 0$: One root is zero, and the other root is $-\frac{b}{a}-\backslash frac\{b\}\{a\}$.
3. $a=ca\; =\; c$: The roots are reciprocal to each other.
4. Signs of a and $c$c:
   • If $a>0a\; >\; 0$ and , or and $c>0c\; >\; 0$, the roots are of opposite signs.
5. Signs of a, b, and $c$c:
   • If $a>0,b>0,c>0a\; >\; 0,\; b\; >\; 0,\; c\; >\; 0$ or , both roots are negative.
   • If or , both roots are positive.
6. Sign Relationship:
   • If the sign of $a$a is the same as the sign of $b$b and different from the sign of $c$c, the greater root in magnitude is negative.
   • If the sign of $b$b is the same as the sign of $c$c and different from the sign of $a$a, the greater root in magnitude is positive.
7. Sum of Coefficients:
   • If $a+b+c=0a\; +\; b\; +\; c\; =\; 0$, one root is 1 and the other root is $\frac{c}{a}\backslash frac\{c\}\{a$ or $\frac{-}{(}\backslash frac\{-(b\; +\; a)\}\{a\}$.

Identity in Equations

An identity is an equation that holds true for every value of the variable within its domain. Examples include:

• $5(a-3)=5a-155(a\; -\; 3)\; =\; 5a\; -\; 15$
• $(a+b)\mathrm{<sup\; style="">2</sup>}=a\mathrm{<sup\; style="">2</sup>}+b\mathrm{<sup\; style="">2</sup>}+2ab(a\; +\; b)^2\; =\; a^2\; +\; b^2\; +\; 2ab$

Quadratic Identity

A quadratic equation cannot have more than two roots. If it seems to have three or more roots, it becomes an identity. For the quadratic equation $ax\mathrm{<sup\; style="">2</sup>}+bx+c=0$ to be an identity, the coefficients must all be zero: $a=b=c=0.a\; =\; b\; =\; c\; =\; 0$

Common Roots of Two Quadratic Equations

One Common Root

Consider two quadratic equations: $ax\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}+bx+c=0ax^2\; +\; bx\; +\; c\; =\; 0$ ,  $a\prime x\mathrm{<sup\; style="">2</sup>}+b\prime x+c\prime =0.$

If they share a common root $\alpha \backslash alph$, then: $a\alpha \mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}+b\alpha +c=0a\backslash alpha^2\; +\; b\backslash alpha\; +\; c\; =\; 0$,  $a\prime \alpha \mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}+b\prime \alpha +c\prime =0.a\text{'}\backslash alpha^2\; +\; b\text{'}\backslash alpha\; +\; c\text{'}\; =\; 0$

Using Cramer’s Rule to solve for 

The condition for a common root is: $(ca\prime -c\prime a)2=(ab\prime -a\prime b)(bc\prime -b\prime c)(ca\text{'}\; -\; c\text{'}a)^2\; =\; (ab\text{'}\; -\; a\text{'}b)(bc\text{'}\; -\; b\text{'}c)$

Both Roots the Same

If both roots are the same for the two equations, then: 

Remainder Theorem

When dividing a polynomial $f\left(x\right)f(x)$f(x) by $(x-a)(x\; -\; a)$(x−a), the remainder is $f\left(a\right)f(a)$f(a). If $f\left(a\right)=0f(a)\; =\; 0$f(a)=0, then $(x-a)(x\; -\; a)$(x−a) is a factor of $f\left(x\right)f(x)$f(x).

Example

Consider the polynomial: $f\left(x\right)=x3-9x2+23x-15f(x)\; =\; x^3\; -\; 9x^2\; +\; 23x\; -\; 15$f(x)=x3−9x2+23x−15

If $f\left(1\right)=0f(1)\; =\; 0$f(1)=0, then $(x-1)(x\; -\; 1)$(x−1) is a factor of $f\left(x\right)f(x)$f(x).

For another example: $f\left(x\right)=(x-2)(x2-7x+9)+3f(x)\; =\; (x\; -\; 2)(x^2\; -\; 7x\; +\; 9)\; +\; 3$f(x)=(x−2)(x2−7x+9)+3

Here, $f\left(2\right)=3f(2)\; =\; 3$f(2)=3 is the remainder when $f\left(x\right)f(x)$f(x) is divided by $(x-2)(x\; -\; 2)$(x−2).

Rational Inequalities

Given an expression $\frac{f}{\left(}=y\backslash frac\{f(x)\}\{g(x)\}\; =\; y$ where $f\left(x\right)f(x)$f(x) and $g\left(x\right)g(x)$g(x) are polynomials in $xx$, to solve $y>0y\; >\; 0$ (or ), follow these steps:

Factorize the Polynomials: Write $f\left(x\right)f(x)$ and $g\left(x\right)g(x)$ in their factored forms:

$f\left(x\right)=(x-a\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">1</sub>})\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">n</sup>}\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">1</sup>}(x-a\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sub>})\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">n</sup>}\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}\cdots (x-a\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">k</sub>})\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">n</sup>}\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">k</sup>}f(x)\; =\; (x\; -\; a\_1)^\{n\_1\}(x\; -\; a\_2)^\{n\_2\}\; \backslash cdots\; (x\; -\; a\_k)^\{n\_k\}$  

$g\left(x\right)=(x-b\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">1</sub>})\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">m</sup>}\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">1</sup>}(x-b\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sub>})\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">m</sup>}\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}\cdots (x-b\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">p</sub>})\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">m</sup>}\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">p</sup>. }g(x)\; =\; (x\; -\; b\_1)^\{m\_1\}(x\; -\; b\_2)^\{m\_2\}\; \backslash cdots\; (x\; -\; b\_p)^\{m\_p\}$

Here, $a\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">1</sub>},a\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sub>},\dots ,a\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">k</sub>}a\_1,\; a\_2,\; \backslash ldots,\; a\_k$ are the roots of $f\left(x\right)=0f(x)\; =\; 0$ and $b\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">1</sub>},b\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sub>},\dots ,b\mathrm{<sub\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">p</sub>}$ are the roots of $g\left(x\right)=0g(x)\; =\; 0$

1. Marking Zeros and Points of Discontinuity: On a number line, mark the zeros of $f\left(x\right)f(x)$f(x) with black dots (since they make $y=0y\; =\; 0$y=0) and the zeros of $g\left(x\right)g(x)$g(x) with white dots (since they make $yy$y undefined).

   Example:

   

   Marks:  $-6,-2,0,1,3,7-6,\; -2,\; 0,\; 1,\; 3,\; 7$−6,−2,0,1,3,7

2. Testing the Sign: Check $yy$ for any real number greater than the rightmost marked point. If positive, $yy$ is positive for all numbers greater than this point, and vice versa.

3. Identifying Simple and Double Points:

   • If the exponent of a factor is odd, the point is a simple point.
   • If the exponent of a factor is even, the point is a double point.

4. Drawing the Wavy Curve: Starting from the rightmost point, draw a wavy curve above (if $yy$y is positive) or below (if $yy$y is negative) the number line. Cross the number line at simple points and touch but do not cross at double points.

   Example:

   

   Simple points: $1,3,-6,71,\; 3,\; -6,\; 7$1,3,−6,7  ,  Double points: $-2,0-2,\; 0$−2,0
   Draw the curve accordingly.

5. Determining Intervals: The intervals where the curve is above the number line indicate $y>0y\; >\; 0$. The intervals below indicate . Choose the intervals according to the inequality sign.

Important Notes:

1. Denominator Zeros Exclusion: Points where the denominator is zero are never included in the solution.

2. Non-negative/Non-positive Intervals: If asked for non-negative or non-positive intervals, make the intervals closed for roots of the numerator and open for roots of the denominator.

3. Cross-Multiplication: Normally not allowed, but permissible if the denominator is always positive.

4. Squaring in Inequalities: Generally not allowed, unless both sides are non-negative.

5. Multiplication by Negative Number: Allowed with a change in the inequality sign.

6. Addition or Subtraction: Allowed without changing the inequality sign.

Example Problems:

1. Find $x$ such that : Since , the quadratic expression is always positive. Hence, no solution.

2. Solve $(x2-x-6)(x2+6x)>0(x^2\; -\; x\; -\; 6)(x^2\; +\; 6x)\; >\; 0$: Factorize:

   $(x-3)(x+2)x(x+6)>0(x-3)(x+2)x(x+6)\; >\; 0$

   Mark zeros on the number line: $-6,-2,0,3-6,\; -2,\; 0,\; 3$−6,−2,0,3.

   From the graph:

   $x\in (-\infty ,-6)\cup (-2,0)\cup (3,\infty )x\; \backslash in\; (-\backslash infty,\; -6)\; \backslash cup\; (-2,\; 0)\; \backslash cup\; (3,\; \backslash infty)$

Quadratic Expression and Its Graph

Consider the quadratic expression $y=ax2+bx+cy\; =\; ax^2\; +\; bx\; +\; c$y=ax2+bx+c, $a\ne 0a\; \backslash neq\; 0$a=0.

• Graph Shape:

  • $a>0a\; >\; 0$: Parabola opens upwards (concave up).
  • : Parabola opens downwards (concave down).

• Graph Categories:

  • Two real roots $aa$ and $bb$ where :

    $ax\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}+bx+c>0forx\in (-\infty ,a)\cup (b,\infty )ax^2\; +\; bx\; +\; c\; >\; 0\; \backslash quad\; \backslash text\{for\}\; \backslash quad\; x\; \backslash in\; (-\backslash infty,\; a)\; \backslash cup\; (b,\; \backslash infty)$,  

  • One real root $a=ba\; =\; b$:

    $ax\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}+bx+c=0forx=\mathrm{a.}ax^2\; +\; bx\; +\; c\; =\; 0\; \backslash quad\; \backslash text\{for\}\; \backslash quad\; x\; =$

  • No real roots:

    $ax\mathrm{<sup\; style="">2</sup>}+bx+c>0for all\mathrm{ \; x}\in R(ifa>0)ax^2\; +\; bx\; +\; c\; >\; 0\; \backslash quad\; \backslash text\{for\; all\}\; \backslash quad\; x\; \backslash in\; \backslash mathbb\{R\}\; \backslash quad\; (\backslash text\{if\; \}\; a\; >\; 0)$  , .

Important Notes:

1. Positive Quadratic Expression:

2. Negative Quadratic Expression:

Maximum and Minimum Values:

For $y=ax\mathrm{<sup\; style="word-break:\; break-word;\; line-height:\; 1.8;\; color:\; black;\; letter-spacing:\; inherit\; !important;">2</sup>}+bx+cy\; =\; ax^2\; +\; bx\; +\; c$:

• Minimum value at vertex when $a>0a\; >\; 0$:

  $x\; =\; -\backslash frac\{b\}\{2a\},\; \backslash quad\; y\; =\; -\backslash frac\{D\}\{4a\}$

• Maximum value at vertex when :

  $\mathrm{<img\; src="https://edurev.gumlet.io/cdn\_assets/v321/assets/img/ph.png"\; style="display:\; block;\; margin:\; 5px\; auto;\; text-align:\; center;\; word-break:\; break-word;\; width:\; auto;\; max-width:\; 100\%;\; letter-spacing:\; inherit\; !important;"\; fr-original-class="fr-draggable"\; alt="Notes:\; Quadratic\; Equations\; |\; Mathematics\; (Maths)\; for\; JEE\; Main\; \&\; Advanced"\; data-src="https://edurev.gumlet.io/ApplicationImages/Temp/20313831\_08e3c416-5fbd-4110-85f9-eb678b35dc2e\_lg.png">}$

$x^2\; -\; \backslash left(\; -\backslash frac\{b\}\{a\}\; \backslash right)x\; +\; \backslash frac\{c\}\{a\}\; =\; 0$