Which of the following is a quadratic equation?

x^2 - 5x + 6 = 0. A quadratic equation must have the highest power of the variable as 2, and no variable in the denominator.

For the equation x^2 - 4x + 4 = 0, what is the value of the discriminant?

0. The discriminant is b^2 - 4ac. Here a=1, b=-4, c=4. So, (-4)^2 - 4(1)(4) = 16 - 16 = 0.

If the discriminant of a quadratic equation is negative, what can be said about its roots?

There are no real roots (complex roots).. A negative discriminant means taking the square root of a negative number, which results in complex (non-real) roots.

Mathematics · Class 9-10 · CBSE

Quadratic Equations

Aquadratic equationis a polynomial equation of the second degree, meaning it contains at least one term where the variable is squared. These equations are fundamental in mathematics and have wide applications in science and engineering.

1What is a Quadratic Equation?

A quadratic equation in one variable is an equation that can be written in the standard form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are real numbers and $a \neq 0$ . The highest power of the variable $x$ is 2.

Components of a Quadratic Equation

Component	Description	Example
$ax^2$	The quadratic term (coefficient $a \neq 0$ )	$3x^2$
$bx$	The linear term (coefficient $b$ )	$5x$
$c$	The constant term	$-2$

$ax^2$
Description
The quadratic term (coefficient $a \neq 0$ )
Example
$3x^2$
$bx$
Description
The linear term (coefficient $b$ )
Example
$5x$
$c$
Description
The constant term
Example
$-2$

2Methods to Solve Quadratic Equations

There are several methods to find the values of $x$ that satisfy a quadratic equation, also known as itsrootsorsolutions. The most common methods are factoring, completing the square, and using the quadratic formula.

Solving by Factoring

1
Standard Form
Rewrite equation as $ax^2 + bx + c = 0$
2
Factor
Find two numbers that multiply to $ac$ and add to $b$
3
Group Terms
Rewrite $bx$ using these numbers and factor by grouping
4
Zero Product
Set each factor to zero and solve for $x$

3The Quadratic Formula

Thequadratic formulais a universal method to find the roots of any quadratic equation, regardless of whether it can be factored easily. It directly provides the values of $x$ .

Quadratic Formula

x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

$x$ = roots of the equation
$a, b, c$ = coefficients from $ax^2 + bx + c = 0$

Common mistake: A common mistake is forgetting to divide the entire numerator by

2a

, or making sign errors with

-b

and

b^2 - 4ac

4Discriminant and Nature of Roots

The expression under the square root in the quadratic formula, $b^2 - 4ac$ , is called thediscriminant( $\Delta$ ). It tells us about the nature of the roots without actually solving the equation.

5Example: Solving a Quadratic Equation

Real-life example · Solve $2x^2 + 5x - 3 = 0$

Here, $a=2$ , $b=5$ , $c=-3$ . Using the quadratic formula: $x = \frac{-5 \pm \sqrt{5^2 - 4(2)(-3)}}{2(2)}$ . This simplifies to $x = \frac{-5 \pm \sqrt{25 + 24}}{4} = \frac{-5 \pm \sqrt{49}}{4} = \frac{-5 \pm 7}{4}$ . So, $x_1 = \frac{-5 + 7}{4} = \frac{2}{4} = \frac{1}{2}$ and $x_2 = \frac{-5 - 7}{4} = \frac{-12}{4} = -3$ . The roots are $1/2$ and $-3$ .

Exam tip: Always check your solutions by substituting them back into the original equation to ensure they satisfy it.

Remember

Aquadratic equationis a polynomial of degree 2, written as $ax^2 + bx + c = 0$ where $a \neq 0$ .
Therootsor solutions are the values of $x$ that satisfy the equation.
Quadratic equations can be solved byfactoring,completing the square, or using thequadratic formula.
Thequadratic formulais $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
Thediscriminant, $\Delta = b^2 - 4ac$ , determines the nature of the roots (real, distinct, or complex).

Check yourself

Question 1
Which of the following is a quadratic equation?
Question 2
For the equation $x^2 - 4x + 4 = 0$ , what is the value of the discriminant?
Question 3
If the discriminant of a quadratic equation is negative, what can be said about its roots?