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Chapter 4

Quadratic Equations

NCERT solutions and explanations for Class 10 Mathematics Chapter 4 Quadratic Equations — covering Quadratic equations, solution by factorisation, completing the square, quadratic formula, discriminant, and word problems.

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Questions

6
Q1

Define a quadratic equation and identify aa, bb, and cc in 2x2+x−300=02x^2+x-300=0.

A quadratic equation in xx is ax2+bx+c=0ax^2+bx+c=0, where a,b,ca,b,c are real numbers and a≠0a\ne0. In 2x2+x−300=02x^2+x-300=0, a=2a=2, b=1b=1, c=−300c=-300.
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Q2

Solve x2−5x+6=0x^2-5x+6=0 by factorisation.

Factorise: x2−5x+6=(x−2)(x−3)x^2-5x+6=(x-2)(x-3). Therefore (x−2)(x−3)=0(x-2)(x-3)=0, so x=2x=2 or x=3x=3.
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Q3

Explain the completing-the-square idea for solving a quadratic equation.

For a quadratic equation ax2+bx+c=0ax^2+bx+c=0 with a≠0a\ne0, first divide by aa and add and subtract (b2a)2\left(\frac{b}{2a}\right)^2 to form a perfect square. This gives (x+b2a)2=b2−4ac4a2\left(x+\frac{b}{2a}\right)^2=\frac{b^2-4ac}{4a^2}, after which taking square roots gives the solutions.
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Q4

State the quadratic formula and the condition for real roots.

For ax2+bx+c=0ax^2+bx+c=0, a≠0a\ne0, the roots are x=−b±b2−4ac2ax=\frac{-b\pm\sqrt{b^2-4ac}}{2a}. Real roots exist when b2−4ac≥0b^2-4ac\ge0.
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Q5

Find the nature of roots of 2x2−4x+3=02x^2-4x+3=0.

Here a=2a=2, b=−4b=-4, c=3c=3. Discriminant D=b2−4ac=16−24=−8<0D=b^2-4ac=16-24=-8<0. Therefore the equation has no real roots.
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Q6

A rectangular hall has area 300 m2300\text{ m}^2 and length one metre more than twice its breadth. Form the quadratic equation for its breadth.

Let breadth be xx m. Length is (2x+1)(2x+1) m. Area =x(2x+1)=300=x(2x+1)=300, so 2x2+x−300=02x^2+x-300=0.
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