CBSE Class 10 Maths Notes: Quadratic Equations

Standard Form of a Quadratic Equation

A quadratic equation is an equation that can be written in the standard form:
$ax^2 + bx + c = 0$, where $a \ne 0$. Here,

• $x$ is the variable.
• $a$, $b$, and $c$ are real numbers, and $a$ is not equal to zero.
• $a$ is the coefficient of $x^2$, $b$ is the coefficient of $x$, and $c$ is the constant term.

Solving Quadratic Equations – Factorization Method

Definition: Factorization involves expressing the quadratic expression as a product of two linear factors.

Steps for Factorization:

1. Step 1: Ensure the equation is in standard form ($ax^2 + bx + c = 0$).
2. Step 2: Find two numbers whose product is equal to $ac$ and whose sum is equal to $b$.
3. Step 3: Rewrite the middle term ($bx$) using these two numbers.
4. Step 4: Factor by grouping.
5. Step 5: Set each factor equal to zero and solve for $x$ to find the roots (solutions).

Example: Solve $x^2 + 5x + 6 = 0$.

• We need two numbers that multiply to 6 and add to 5. These numbers are 2 and 3.
• Rewrite the equation: $x^2 + 2x + 3x + 6 = 0$.
• Factor by grouping: $x(x+2) + 3(x+2) = 0$.
• $(x+2)(x+3) = 0$.
• Therefore, $x+2 = 0$ or $x+3 = 0$.
• The roots are $x = -2$ and $x = -3$.

Solving Quadratic Equations – Quadratic Formula

Formula: The quadratic formula provides a general solution for any quadratic equation in the form $ax^2 + bx + c = 0$.
Formula: $x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$

Steps for using the Quadratic Formula:

1. Step 1: Identify the coefficients $a$, $b$, and $c$ from the standard form of the equation.
2. Step 2: Substitute the values of $a$, $b$, and $c$ into the quadratic formula.
3. Step 3: Simplify to find the two roots. The $\pm$ symbol indicates that there are two possible solutions: one where you add the square root and one where you subtract it.

Example: Solve $2x^2 + 7x + 3 = 0$.

• Here, $a = 2$, $b = 7$, and $c = 3$.
• Substitute these values into the quadratic formula: $x = \frac{-7 \pm \sqrt{7^2 – 4(2)(3)}}{2(2)}$.
• Simplify: $x = \frac{-7 \pm \sqrt{49 – 24}}{4} = \frac{-7 \pm \sqrt{25}}{4} = \frac{-7 \pm 5}{4}$.
• Therefore, the roots are $x = \frac{-7 + 5}{4} = -\frac{1}{2}$ and $x = \frac{-7 – 5}{4} = -3$.

Discriminant and Nature of Roots

Definition: The discriminant is a part of the quadratic formula, and it tells us about the nature of the roots of a quadratic equation.

Formula: The discriminant, denoted by $D$ (or sometimes $\Delta$), is given by:
$D = b^2 – 4ac$

Nature of Roots Based on the Discriminant:

• If $D > 0$: The equation has two distinct real roots (the graph of the quadratic equation crosses the x-axis at two points).
• If $D = 0$: The equation has two real and equal roots (the graph touches the x-axis at one point).
• Focus on Real Roots: We will not consider the cases where $D<0$ in this context. If $D < 0$, the roots are non-real (complex), and the graph doesn't intersect the x-axis.

Example:

• For $x^2 + 5x + 6 = 0$: $D = 5^2 – 4(1)(6) = 25 – 24 = 1$. Since $D > 0$, the equation has two distinct real roots.
• For $x^2 + 4x + 4 = 0$: $D = 4^2 – 4(1)(4) = 16 – 16 = 0$. Since $D = 0$, the equation has two real and equal roots.

Situational Problems Leading to Quadratic Equations

Many real-world problems can be modeled by quadratic equations. These problems often involve areas, perimeters, speeds, time, and other related concepts.

General Approach to Solving Situational Problems:

1. Step 1: Read the problem carefully and identify the unknown quantity (let it be $x$ or another variable).
2. Step 2: Translate the problem into a quadratic equation by forming expressions based on the information given.
3. Step 3: Solve the quadratic equation (using factorization or the quadratic formula).
4. Step 4: Check your solutions against the context of the problem and discard any solutions that don’t make sense (e.g., negative length or time).

Example: Find two numbers whose sum is 9 and product is 20.

• Let one number be $x$. Then the other number is $9 – x$.
• Their product is $x(9 – x) = 20$.
• This gives us the quadratic equation: $9x – x^2 = 20$ or $x^2 – 9x + 20 = 0$.
• Factoring, we get $(x – 4)(x – 5) = 0$.
• Therefore, $x = 4$ or $x = 5$.
• The two numbers are 4 and 5.