QUADRATIC EQUATIONS

Introduction

A quadratic equation is a second-degree polynomial equation. The defining characteristic of a quadratic equation is the squared variable (e.g., $x^2$). When graphed, a quadratic equation forms a U-shaped curve called a parabola.

Brief Overview

graph TD; Quad[Quadratic ax^2+bx+c=0] --> Factor[Factorisation]; Quad --> Formula[Quadratic Formula]; Formula --> Disc[Discriminant b^2-4ac]; Quad --> Graph[Graph: Parabola];

1. Standard Form and Basics

Standard Form: $ax^2 + bx + c = 0$, where $a, b,$ and $c$ are real numbers, and $a \ne 0$.
- $ax^2$ is the quadratic term.
- $bx$ is the linear term.
- $c$ is the constant term.
Roots/Solutions: The values of $x$ that satisfy the equation. A quadratic equation can have two real roots, one real root (a repeated root), or no real roots (complex roots).

Numerical Example:
Convert $3x(x - 2) = 5$ into standard form.
Expand: $3x^2 - 6x = 5$.
Subtract 5: $3x^2 - 6x - 5 = 0$.
Here, $a=3, b=-6, c=-5$.

MCQs - Standard Form

The highest power of the variable in a quadratic equation is:
a) 1
b) 2
c) 3
d) 4
Answer: b
The standard form of a quadratic equation is:
a) $y = mx + b$
b) $ax^2 + bx + c = 0$
c) $ax + b = 0$
d) $x^2 + y^2 = r^2$
Answer: b
In the equation $5x^2 - 7x + 2 = 0$, the value of 'b' is:
a) 5
b) 7
c) -7
d) 2
Answer: c
Which of the following is NOT a quadratic equation?
a) $x^2 = 9$
b) $2x^2 + 5x = 0$
c) $3x + 4 = x^2$
d) $x^3 + x^2 = 0$
Answer: d
A quadratic equation can have a maximum of how many real roots?
a) 1
b) 2
c) 3
d) Infinite
Answer: b
Which coefficient determines the direction a parabola opens?
a) a
b) b
c) c
d) Both a and b
Answer: a
If $a = 0$ in the equation $ax^2 + bx + c = 0$, the equation becomes:
a) Cubic
b) Linear
c) Constant
d) Undefined
Answer: b
What is the constant term in $4x^2 - 9 = 0$?
a) 4
b) -9
c) 9
d) 0
Answer: b
The coefficient of the linear term in $x^2 + c = 0$ is:
a) 1
b) c
c) 0
d) -1
Answer: c
Which equation is equivalent to $x(x + 2) = 3$?
a) $x^2 + 2x = 3$
b) $x^2 + 2x - 3 = 0$
c) $x^2 - 3 = 0$
d) $x^2 + 2 = 3$
Answer: b

2. Solving by Factorization (Splitting the Middle Term)

Method: This method involves breaking the linear term ($bx$) into two terms whose sum is $b$ and whose product is $a \times c$.
Zero Product Property: If $A \times B = 0$, then either $A = 0$ or $B = 0$. After factoring the quadratic into two binomials, set each to zero to find the roots.

Numerical Example:
Solve $x^2 - 5x + 6 = 0$.
Find two numbers that add to $-5$ and multiply to $6$ (which are $-2$ and $-3$).
Split middle term: $x^2 - 2x - 3x + 6 = 0$.
Factor by grouping: $x(x - 2) - 3(x - 2) = 0$.
$(x - 2)(x - 3) = 0$.
Roots: $x - 2 = 0 \Rightarrow x = 2$; $x - 3 = 0 \Rightarrow x = 3$.

MCQs - Solving by Factorization

To solve by splitting the middle term for $ax^2 + bx + c = 0$, we find two numbers that sum to $b$ and multiply to:
a) $c$
b) $ac$
c) $a+c$
d) $a$
Answer: b
If $(x - 4)(x + 5) = 0$, the roots are:
a) 4 and 5
b) -4 and -5
c) 4 and -5
d) -4 and 5
Answer: c
Factor $x^2 - 9 = 0$. The roots are:
a) 3 only
b) 9 and -9
c) 3 and -3
d) 81
Answer: c (Using difference of squares: $(x-3)(x+3)=0$)
The roots of $x^2 + 6x + 9 = 0$ are:
a) 3 and -3
b) -3 and -3
c) 9 and 1
d) -9 and -1
Answer: b (It factors to $(x+3)^2 = 0$)
The Zero Product Property states that if $ab = 0$, then:
a) $a = b$
b) $a = 1$ or $b = 1$
c) $a = 0$ or $b = 0$
d) $a+b = 0$
Answer: c
If the factors of a quadratic equation are $(2x - 1)$ and $(x + 3)$, the roots are:
a) $1/2$ and $-3$
b) $-1/2$ and $3$
c) $2$ and $-3$
d) $1$ and $3$
Answer: a
What is the factored form of $x^2 - 5x + 6 = 0$?
a) $(x - 1)(x - 6) = 0$
b) $(x - 2)(x - 3) = 0$
c) $(x + 2)(x + 3) = 0$
d) $(x - 3)(x + 2) = 0$
Answer: b
To solve $x^2 - 16 = 0$ by factorization, we use the identity:
a) Perfect square trinomial
b) Difference of two squares
c) Sum of cubes
d) Difference of cubes
Answer: b
The roots of $x^2 = 5x$ are:
a) 5 only
b) 0 and 5
c) -5 and 5
d) 1 and 5
Answer: b
If a quadratic equation factors perfectly as $(x - k)^2 = 0$, the roots are:
a) Real and distinct
b) Real and equal
c) Complex
d) Irrational
Answer: b

3. Solving by the Quadratic Formula

If a quadratic equation cannot be easily factored, the quadratic formula will always work.
Formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
The $\pm$ symbol indicates that there are usually two solutions: one using the $+$ and one using the $-$.

Numerical Example:
Solve $x^2 + 4x + 1 = 0$. Here $a=1, b=4, c=1$.
$x = \frac{-4 \pm \sqrt{4^2 - 4(1)(1)}}{2(1)}$
$x = \frac{-4 \pm \sqrt{16 - 4}}{2} = \frac{-4 \pm \sqrt{12}}{2}$
Since $\sqrt{12} = 2\sqrt{3}$, $x = \frac{-4 \pm 2\sqrt{3}}{2} = -2 \pm \sqrt{3}$.

MCQs - Quadratic Formula

The quadratic formula is:
a) $x = \frac{b \pm \sqrt{b^2 - 4ac}}{2a}$
b) $x = \frac{-b \pm \sqrt{b^2 + 4ac}}{2a}$
c) $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
d) $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{a}$
Answer: c
In the quadratic formula, the expression under the square root is:
a) $b^2 + 4ac$
b) $b^2 - 4ac$
c) $4ac - b^2$
d) $a^2 - 4bc$
Answer: b
The quadratic formula can be used to solve:
a) Only factorable quadratic equations
b) Any quadratic equation
c) Linear equations
d) Cubic equations
Answer: b
Use the formula on $x^2 - 2x - 3 = 0$. What is the value of $-b$?
a) -2
b) 2
c) 3
d) -3
Answer: b
If $a=1, b=0, c=-16$, the formula yields:
a) $x = \pm 4$
b) $x = \pm 16$
c) $x = 0$
d) No solution
Answer: a ($x = \frac{0 \pm \sqrt{0 - 4(1)(-16)}}{2} = \frac{\pm \sqrt{64}}{2} = \pm 4$)
In the quadratic formula, the denominator is:
a) $a$
b) $2a$
c) $b$
d) $2b$
Answer: b
If the quadratic formula results in $\frac{2 \pm 0}{2}$, the root is:
a) 0
b) 1
c) 2
d) Undefined
Answer: b
For the equation $2x^2 - x - 1 = 0$, what are the values of $a, b, c$?
a) $2, 1, 1$
b) $2, -1, 1$
c) $2, -1, -1$
d) $-2, 1, 1$
Answer: c
The quadratic formula is derived from which method?
a) Factorization
b) Completing the square
c) Graphical method
d) Synthetic division
Answer: b
Applying the formula to $x^2 + 1 = 0$ yields:
a) Real roots
b) $x = \pm 1$
c) $x = \pm i$ (no real roots)
d) $x = 0$
Answer: c

4. Nature of Roots (The Discriminant)

Discriminant ($\Delta$): The part of the quadratic formula under the square root: $D = b^2 - 4ac$. It determines the "nature" of the roots without actually solving the equation.
If $D > 0$: There are two distinct, real roots.
If $D = 0$: There is exactly one real root (or two equal real roots).
If $D < 0$: There are no real roots (the roots are imaginary/complex).

Numerical Example:
Find the nature of the roots of $2x^2 - 3x + 5 = 0$.
$D = b^2 - 4ac = (-3)^2 - 4(2)(5) = 9 - 40 = -31$.
Since $D < 0$, the equation has no real roots.

MCQs - Nature of Roots

The discriminant of a quadratic equation is denoted by:
a) $ax^2$
b) $-b/2a$
c) $b^2 - 4ac$
d) $\sqrt{ac}$
Answer: c
If the discriminant $D > 0$, the quadratic equation has:
a) No real roots
b) Two distinct real roots
c) One repeated real root
d) Infinite roots
Answer: b
For the roots to be real and equal, the discriminant must be:
a) Greater than 0
b) Less than 0
c) Exactly 0
d) A perfect square
Answer: c
What is the discriminant of $x^2 - 4x + 4 = 0$?
a) 16
b) 0
c) -16
d) 8
Answer: b ($(-4)^2 - 4(1)(4) = 16 - 16 = 0$)
A quadratic equation whose graph (parabola) does not touch the x-axis at all has a discriminant that is:
a) Positive
b) Zero
c) Negative
d) 1
Answer: c (No real roots means no x-intercepts)
If $b^2 - 4ac = 0$, the parabola touches the x-axis at exactly:
a) Zero points
b) One point
c) Two points
d) Three points
Answer: b
For the equation $3x^2 + x - 1 = 0$, the discriminant is:
a) 13
b) -11
c) 11
d) -13
Answer: a
A negative discriminant indicates that the roots are:
a) Rational
b) Irrational
c) Real and distinct
d) Complex (imaginary)
Answer: d
If $D > 0$ and $D$ is a perfect square, the roots are:
a) Rational
b) Irrational
c) Equal
d) Imaginary
Answer: a
If the roots are real and distinct, then:
a) $D < 0$
b) $D = 0$
c) $D > 0$
d) $D \le 0$
Answer: c

Fun Facts about Quadratic Equations!

Parabolas in Real Life: The path of any projectile—like a thrown basketball, an angry bird, or a cannonball—follows the shape of a parabola, which is perfectly modeled by a quadratic equation!
Babylonian Math: The Babylonians were solving quadratic equations over 4,000 years ago using completing the square, though they didn't have the modern formula or even zero!
Golden Ratio: The famous Golden Ratio ($\phi \approx 1.618$) is actually a root of the simple quadratic equation $x^2 - x - 1 = 0$.