## Quadratic Equations Maths Notes

**Revision:**

A polynomial in one variable having degree 1 is called a linear polynomial, and having degree 2 is called a quadratic polynomial.

**Activity:**

Classify the following polynomials into linear polynomials and quadratic polynomials :

5x + 9, x^{2} + 3x – 5, 3x – 7, 3x^{2} – 5x, 5x^{2}

The equation obtained by taking the value of a quadratic polynomial zero, is called a quadratic equation.

x^{2} + 3x – 5 = 0; 3x^{2} – 5x = 0; 5x^{2} = 0 are quadratic equations formed by taking the value of the quadratic polynomials, in the above box, zero.

**Activity:**

x^{2} + 3x – 5, 3x^{2} – 5x, 5x^{2}. Write the polynomials in the index form.

x^{2} + 3x – 5, 3x^{2} – 5x + 0, 5x^{2} + 0x + 0

Coefficients of x^{2} are 1, 3, and 5. These coefficients are non-zero.

Coefficients of x are 3, – 5 and 0 respectively.

Constants terms are – 5, 0 and 0 respectively. Here constant terms of second and third polynomials are zero.

**Standard Form of the Quadratic Equation:**

The equation involving one variable and having the maximum index of the variable 2 is called a quadratic equation.

- The equation ax
^{2}+ bx + c = 0 is called the standard form of quadratic equation. Here, a, b and c are real numbers and a ≠ 0. - In the equation ax
^{2}+ bx + c = 0, if b = 0, then the equation becomes ax^{2}+ c = 0. This is also a quadratic equation. - Similarly, if c = 0, then ax
^{2}+ bx = 0 is a quadratic equation. - If b = 0 and c = 0, then ax
^{2}= 0 is a quadratic equation.

In the quadratic equation ax^{2}+ bx + c = 0, the constants a, b, c are called the quadratic coefficient, the linear coefficient and the constant term respectively.

**Activity: Complete the following table:**

Decide which of the following are quadratic equations.

**Roots of the Quadratic Equation:**

The values of the variable which satisfy the given quadratic equation are called the roots of the quadratic equation.

Activity:

If x = 5 is a root of the equation kx^{2} – 14x – 5 = 0 then find the value of k by completing the following activity.

One of the roots of the equation kx^{2} – 14x – 5 = 0

**Let’s Remember:**

(1) ax^{2} + bx + c = 0 is the standard form of the quadratic equation, where a, b, care real numbers and a is non-zero.

(2) The values of the variable which satisfy the equation are called solutions or the roots of the equation.

**Activity: Factorize the following polynomials:**

**Methods of Solving Quadratic Equations:**

Following are the methods for solving quadratic equations:

- Factorisation method
- Completing square method
- Formula method.

**Solution of a Quadratic Equation by Factorisation**

- Write the given equation in the form ax
^{2}+ bx + c = 0. - Find the two linear factors of the LHS of the equation.
- Equate each linear factor to zero.
- Solve each equation obtained in 3 and write the roots of the given equation.

**Study the following example:**

Solve the quadratic equation x^{2} + 8x + 15 = 0 by factorisation method:

Steps | Quadratic Equation: x^{2} + 8x + 15 =0 |

1. Split the middle term 8x as 3x and 5x. [Because 3x × 5x = 15x^{2} = x^{2} × 15.] |
x^{2} + 3x + 5x + 15 = 0 |

2. Find the factors of LHS | x(x +3) + 5(x +3) = 0 (x + 3)(x + 5) = 0 |

3. If the product of two numbers is zero, then at least one of them must be zero. | x + 3 = 0 or x + 5 = 0 |

4. Solve each linear equation. | x = – 3 or x = – 5 |

5. Write the answer. | The roots of the equation are – 3, – 5. |

**Solution of a Quadratic Equation by Completing Square Method:**

The quadratic equation of the type x^{2} + 6x +2 = 0 cannot be solved by the method of factorisation, because we cannot find two factors of 2 whose sum is 6.

In such a case, completing square method is used for solving quadratic equations.

For solving quadratic equation by this method proceed as follows:

(1) Write the given equation in the form ax^{2} + bx + c = 0 |
x^{2} + 6x + 2 = 0 |

(2) Considering the first two terms on LHS, find the third suitable square term to make the polynomial a perfect square. | Comparing x^{2} + 6x with x^{2} + 2xy, 2xy = 6x∴ y = 3 ∴ y ^{2} = 9x ^{2} + 6x + 9 is a perfect square polynomial. |

(3) Add the square term and subtract the same. | x^{2} + 6x + 2 = 0∴ x ^{2} + 6x + 9 – 9 + 2 = 0 |

(4) Write the square of the first three terms and the last two terms. | (x + 3)^{2} – (√7)^{2} = 0 |

(5) Factorise and equate each factor to zero. | (x + 3 + √7) (x + 3 – √7) = 0 x + 3 + √7 = 0 or x + 3 – √7 = 0 |

(6) Find the value of x. | x = – 3 – √7 or x = – 3 + √7 |

– 3 – √7, – 3 + √7 are the roots of the quadratic equation.

**Formula for Solving a Quadratic Equation**

ax^{2} + bx + c. Divide the polynomial by a(∵a ≠ 0) to get x^{2} + \(\frac{b}{a}\)x + \(\frac{c}{a}\).

Let us write the polynomial x^{2} + \(\frac{b}{a}\)x + \(\frac{c}{a}\) in the form of difference of two square numbers. Now we can obtain roots or solutions of equation x^{2} + \(\frac{b}{a}\)x + \(\frac{c}{a}\) = 0

which is equivalent to ax^{2} + bx + c = 0.

ax^{2} + bx + c = 0

**For more information:**

Solve the quadratic equation x^{2} – x – 6 = 0 graphically.

x^{2} – x – 6 = 0

∴ x^{2} = x + 6.

Let, y = x^{2} = x + 6.

We draw the graphs of y = x^{2} and y = x + 6

The graph of y = x^{2} is a parabola.

The graphs of y = x^{2} and y = x + 6 intersect each other at (3, 9) and (-2, 4).

∴ x = 3 or x = – 2 is the solution of the given quadratic equation.

**Activity 1:**

Write the quadratic equation having sum of the roots 10 and the product of the roots 9.

The quadratic equation :

**Activity 2:**

What is the quadratic equation having the root α = 2 and β = 5?

**Let’s Remember:**

1. If α and β are the roots of the quadratic equation ax^{2} + bx + c = 0, then

2. The nature of the roots of the quadratic equation ax^{2} + bx + c = 0 depends on the value of b^{2} – 4ac. b^{2} – 4ac is called the discriminant and is denoted by ∆(the Greek letter)

3. If ∆ = 0, the roots are real and equal.

If ∆ > 0, the roots are real and unequal.

If ∆ < 0, the roots are not real.

4. The quadratic equation whose roots are α and α is x^{2} – (α + β)x + αβ = 0.

**Application of Quratic Equation**:

Quadratic equations are useful to solve problems arising in our day-to-day life.

The method of solving problems consists of the following three steps:

Step 1: Convert the word problem, into symbolic language, i.e. form mathematical equation by Identifying the relationship existing in the problem.

Step 2: Solve the quadratic equation thus formed.

Step 3: Interpret the solution of the equation Into verbal language. The appropriate solution/solutions satisfying the given conditions is/are to be considered.