Quadratic Equations in one variable

Important Questions

Quadratic equations are a fundamental topic in algebra that deals with equations of the second degree, where the highest power of the variable is squared. In the context of ICSE Class 10 mathematics, quadratic equations in one variable are expressions of the form:

\(ax^2+bx+c=0\)

Where:

1. a, b, and c are constants, with a ≠ 0

2. x is the variable we're solving for

3. \(ax^2\) represents the quadratic term.

4. bx represents the linear term.

5. c is the constant term.

6. The equation is set equal to zero.

These quadratic equation class 10 ICSE important questions serve as invaluable practice tools for upcoming class tests, unit assessments, and examinations. They are strategically designed to bolster students' confidence in their mathematical prowess and address any uncertainties or challenges they may encounter while studying this topic.

Tables of Content

Understanding quadratic equation class 10 ICSE is essential for building a strong foundation in algebra and problem-solving. It equips students with the tools to analyse and solve a wide range of mathematical and practical problems. In your Class 10 quadratic equation in one variable, you will explore different methods to solve quadratic equations, understand the significance of the discriminant, and apply these concepts to real-life situations. This knowledge not only helps in achieving success in mathematics but also develops critical thinking and analytical skills that are valuable in many aspects of life.

Key Concepts:

Solving Quadratic Equations: Solving a quadratic equation means finding the values of x that make the equation true. Various methods are available for solving quadratic equations, including factoring, using the quadratic formula, and completing the square.

Roots of a Quadratic Equation: The solutions to a quadratic equation (values of x that satisfy the equation) are called its roots or solutions. A quadratic equation can have two real roots, one real root (a repeated root), or two complex roots (if the discriminant is negative).

Discriminant: The discriminant (Δ) is a crucial part of the quadratic formula and determines the nature of the roots:

Δ>0: Two distinct real roots.

Δ=0: One real root (repeated).

Δ<0: Two complex roots.

Applications: Quadratic equations are widely used to model various real-world scenarios, including physics (projectile motion), engineering (optimization), finance (interest calculations), and more.

Quadratic equations in one variable as we study in quadratic equation class 10 ICSE are mathematical expressions of the form:

\(ax^2 +bx+c=0\)

Where:

1. a, b, and c are constants, with a ≠ 0

2. x is the variable we're solving for

3. \(ax^2\) represents the quadratic term.

4. bx represents the linear term.

5. c is the constant term.

6. The equation is set equal to zero.

Key concepts related to quadratic equations in one variable include:

Solving Quadratic Equations: Solving a quadratic equation means finding the values of x that satisfy the equation, making it true. Various methods can be used to solve quadratic equations, including factoring, using the quadratic formula, and completing the square.

Roots of a Quadratic Equation: The solutions to a quadratic equation (values of x that satisfy the equation) are called its roots or solutions. A quadratic equation can have two real roots, one real root (a repeated root), or two complex roots (if the discriminant is negative).

Discriminant: The discriminant (Δ) is a crucial part of the quadratic formula and determines the nature of the roots:

Δ>0: Two distinct real roots.

Δ=0: One real root (repeated).

Δ<0: Two complex roots.

Applications: Quadratic equations are widely used to model various real-world scenarios, including physics (projectile motion), engineering (optimization), finance (interest calculations), and more.

\(\frac{1}{2}\)

is a root of the quadratic equation

\(x^2-mx-\frac{5}{4}=0\)

then the value of m is:

(b) -2

(c) -3

(d) 3

**Ans**. (b)-2

**Explanation:**Given

\(x=\frac{1}{2}\)

as root of the equation

\(x^2-mx-\frac{5}{4}=0\\

\therefore \begin{pmatrix}\frac{1}{2} \end{pmatrix}^2-m \begin{pmatrix}\frac{1}{2} \end{pmatrix}-\frac{5}{4}=0\\

\Rightarrow \frac{1}{4}-\frac{m}{2}-\frac{5}{4}=0\)

m = - 2

(b) -2

(c) 1

(d) 2

**Ans**. (a) -1

**Explanation:**\2kx + (2a + b)x – ab = 0

\(\therefore\) a is a root of this equation

\\ 2ka + 2a + ab – ab = 0

\\ 2a(k + 1) = 0

\(\therefore k = –1\)

**Explanation:**Given the equation is,

\4x– 5x – 3 = 0

Compared with ax+ bx + c = 0, we have,

a = 4, b = – 5, c = – 3

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

=\frac{-(-5)+\sqrt{(-5)^2-4×4(-3)}}{2×4}\\

=\frac{5\pm\sqrt{25+48}}{8}=\frac{5\pm\sqrt{73}}{8}=\frac{5\pm8.544}{8}\\

=\frac{5\pm8.544}{8}or \frac{5-8.544}{8}\\

=\frac{13.544}{8}or\frac{-3.544}{8}\)

= 1.693 or – 0.443

= 1.69 or – 0.44 (correct to 2 decimal places) Ans.

**Explanation:**Given equation is, x

⇒ x

On comparing the equation with ax2 + bx + c = 0, we get

∴ a = 1, b = – 3, c = – 9

b

= 9 + 36

= 45

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

= \(\frac{-(-3)\pm\sqrt{45}}{2×1}\)

=\(\frac{3\pm3\sqrt{5}}{2}\)

x =\(\frac{3+3×2.236}{2}\) and \(\frac{3-3×2.236}{2}\)

x =\(\frac{3+6.708}{2}\)and\(\frac{3+6.708}{2}\)

x =\(\frac{9.708}{2}\)and\(\frac{3.708}{2}\)

x = 4·854 and x = – 1·854

∴ x = 4·9 and x = – 1·9 Ans.

**Explanation:** et ABCD be the rectangular garden and CD acts as his house compound wall

Let Length AB = x m

Breadth BC = y m

According to the question,

xy = 200 ...(i)

x + 2y = 50 ...(ii)

From equation (ii), we get

x = 50 – 2y

Putting the value of x in equation (i), we get

(50 – 2y)y = 200

⇒ 50y – 2y

⇒ 2y

⇒y

⇒ y

⇒ y(y – 20) – 5(y – 20) = 0

⇒ (y – 20)(y – 5) = 0

So, y = 5, 20

When y = 5, then

x = 50 – 2y

x = 50 – 2 × 5 = 40

When y = 20, then

x = 50 – 2y

= 50 – 2 × 20 = 10

So, dimensions of the garden are 10 m and 20 m or 40 m and 5 m. Ans.

Class 10 Mathematics, Quadratic Equations in one variable are like special puzzles we solve to find hidden treasures. These equations look like ax2 +bx+c=0 Where a, b, and c are constants, with a ≠ 0 x is the treasure map. So, as you explore quadratic equations in Class 10, think of them as your treasure hunt, your secret code, and your path to becoming a math detective who can uncover hidden gems in the world of mathematics and beyond. For those looking to enhance their skills and develop a more profound understanding of the concepts discussed in this chapter, oswal.io offers an extensive array of practice questions. These quadratic equation class 10 icse board questions have been specifically crafted to support your pursuit of a deeper comprehension of the subject matter.

**Ans: **A quadratic equation in one variable is a polynomial equation of the form ax2 +bx+c=0, where a, b, and c are constants, and x is the variable.

**Ans: ** The roots of a quadratic equation are the values of x that satisfy the equation and make it equal to zero. You can find the roots by factoring, using the quadratic formula, or completing the square.

**Ans: **Yes, a quadratic equation can have no real roots if its discriminant is negative. In such cases, it has two complex roots.

**Ans:** If the coefficient a is positive, the parabola opens upward, and the quadratic equation has a minimum value. The graph is U-shaped.

**Ans:** Yes, some common factorization patterns, such as the difference of squares or perfect squares, can be used as shortcuts to factorize quadratic equations efficiently.